Are constructivists in favor of the Property of Baire? Since the Axiom of Choice proves the law of excluded middle, constructivists reject it. Does this mean that constructivists are in favor of the negation of Choice?
I take this further: Perhaps constructivists are in favor of an axiomatic system stronger than $\mathsf{ZF \neg C}$. Since this system is alone "too negative", I speculate that, constructivists are in favor of accepting a "positive" axiom that clashes with Choice, such as the Axiom of Determinacy or the Property of Baire.
So do the majority of constructivists actually do this? Personally I am not currently a constructivist, but I would be a fan of the Property of Baire if I had to convert to constructivism.
 A: tl;dr In the first section I explain why constructive rejection of choice would not necessarily mean being in favor of negating it. Sections II and III talk about the status of choice itself, and why many constructivists prefer not to take even positive anti-Choice axioms. Finally, Section IV provides a partial answer the question you probably intended to ask (the one about anti-Choice axioms in mathematics without LEM in general, rather than the philosophical constructivism)
I. Constructivists are a diverse bunch, not cogs in some kind of unified political movement. There are no chief whips to tell you whether you can or cannot use various axioms. At best, there are some identifiable schools whose adherents reliably take various axioms (such as the Russian school of constructive mathematics, associated with Markov's principle; or Intuitionism, associated with Brouwer's fan theorem), but not all constructivists belong to any of these.
Moreover, not many people who work primarily (or even exclusively!) in constructive mathematics have constructivist convictions. Most of the highest-profile contributors to constructive mathematics advocate some form of mathematical pluralism [1].
At this point, constructive mathematics usually just refers to mathematics done without the principle of excluded middle, and without other principles, such as the full axiom of choice, that would imply it. Constructivists themselves might decide to do constructive mathematics, usually for some kind of philosophical reason [2].
Accordingly, the fact that constructivists are forced to reject the Axiom of Choice in no way implies that they'd all be in favor of its negation. Certainly, constructivists universally reject e.g. excluded middle, but that does not mean they are in favor of negating it! (See also the Denial section of Bauer's Five Stages essay.)

II. There's more than meets the eye about "Choice implies Excluded Middle", though.
Consider the following principle: "if $\forall x \in A. \exists y \in B. \varphi(x,y)$ holds, then we  have a function $f: A \rightarrow B$ so that $\forall x \in A. \varphi(x,f(x))$ holds". It looks exactly like the Axiom of Choice ruled out by Diaconescu's theorem (take $\varphi(x,y)$ to denote $y\in x$). Nonetheless, the principle is validated by the Brouwer-Heyting-Kolmogorov interpretation, and is in fact a fully provable theorem of Martin-Löf's Constructive Type Theory!
How do we explain this? How can the status of this principle be so so different in constructive type theory and constructive set theories?! Well, Diaconescu's proof uses multiple principles about sets and set-formation, not just the Axiom of Choice in isolation. Accordingly, the interaction of all the principles used in the proof lead to Excluded Middle, not any one of them. We can put the blame on Choice and reject it, or we can choose to weaken some other principle to prevent the proof from going through. E.g. a more thorough analysis might identify set theory's ability to form unrestricted quotients as the culprit, or in some cases even instances of separation). The former leads to the observation that "extensional Choice" or "setoid Choice" does imply Excluded Middle, even in type theory. The introduction of propositional truncations (which might itself be a subject of some controversy in some schools) and Homotopy Type Theory allows ever more stringent analyses about what sorts of choice principles should be admissible -- and indeed endless debates about which of these principles even deserve the name Axiom of Choice. The Coq Choice facts page provides a good, referenced summary of Choice principles in type theory.
At this point one should mention the fact that some forms of Choice tend to have quite tame consequences over constructive mathematics, even when they have wild consequences over a corresponding classical system. Adding the Axiom of Choice in the form $(\forall x \in \mathbb{N}. \exists y \in \mathbb{N}. \varphi(x,y)) \rightarrow \exists f: \mathbb{N} \rightarrow \mathbb{N}. \forall x \in \mathbb{N}. \varphi(x,f(x))$ to higher-order Peano Arithmetic yields a far more powerful system. Adding it to higher-order Heyting arithmetic has no arithmetic consequences at all.

III. So, how often do constructivists take positive axioms that altogether clash with Choice? In my experience, not that often.
First of all, the Brouwer-Heyting-Kolmogorov interpretation fully justifies Choice: if you practice constructivism because you find the BHK interpretation compelling in some sense, it makes no sense to take axioms that contradict it.
Second of all, a constructivist usually sacrifices Excluded Middle not on a whim, but to achieve some higher constructive goals. Goals such as the existence property, that from a proof of $\exists x. \varphi(x)$ you can without creative work extract some kind of object $t$ that actually satisfies $\varphi(t)$ [3]. Similarly, the disjunction property, which states that from a proof of $A \vee B$ you can without creative work extract either a proof of $A$ or a proof of $B$. These both fail in classical systems.
When the constructivist sacrifices Excluded Middle to obtain a constructive system obeying such desirable properties, it wouldn't really make sense to then adopt a new principle which breaks these properties. This rules out Determinacy, which asserts the existence of winning strategies for various games, but does not provide any methods for finding these strategies (hence breaking the existence property), or even the identity of the winner (hence breaking the disjunction property).
Even universalizing principles, which state that the only examples of some phenomenon are the nice, constructible ones, may in fact seem undesirable from this perspective! Consider the functor $\mathrm{Hom}(\mathbb{N}, -): \mathrm{FinSet} \rightarrow \mathrm{Set}$ and the inclusion functor $\iota: \mathrm{FinSet} \rightarrow \mathrm{Set}$. I once proposed (a more general form of) the following axiom, convenient for category theory: $\mathrm{Nat}(\mathrm{Hom}(\mathbb{N}, -), \iota) \cong \mathbb{N}$. This would constitute a straightforward generalization of the Yoneda lemma, which gives that for a finite set $F$, the natural transformations $\mathrm{Hom}(F, -)$ to the inclusion functor $\iota$ correspond to elements of $F$. It's a positive principle, and not compatible with full Choice since it implies the non-existence of nonprincipal ultrafilters. Moreover, it's easy to see that the only natural transformations from $\mathrm{Hom}(\mathbb{N}, -)$ to $\iota$ that you can actually construct are in fact the evaluation maps at some natural number. And yet this principle is still non-constructive: if you hand me a natural transformation from $\mathrm{Hom}(\mathbb{N}-)$ to $\iota$, I won't be able to constructively recognize which natural number you used to construct it!
The late Troelstra's Uniformity Principle also has a nice topological/constructive universalizing motivation, and breaks LEM (and Choice) in various satisfying ways when added to constructive set theory. Yet it also fails to retain constructive character, similarly to my Yoneda-type axiom above.
It's also worth mentioning that positive anti-Choice principles tend to give you some Choice -- even classically, where Determinacy implies Countable Choice, although it contradicts full Choice. In constructive set theories, you would not want to admit principles which constructively contradict full Choice, but give you some Choice: a little Choice goes a long way toward Excluded Middle.

IV. If you give up on retaining the constructive character of your system, you can add several principles to a constructive set theory (or a topos) that contradict Excluded Middle. The Kock-Lawvere axiom in smooth topoi is a famous example: it contradicts the theorem of classical algebra stating that fields lack zero-divisors. This is the domain of pluralism, more so than constructivism (although I wouldn't rule out a constructive interpretation of smooth topoi one day).
But even in Boolean topoi, failures of Choice are useful: see about $G$-$\mathrm{Set}$ here. Alas, I personally don't know any "mathematically natural" Boolean topoi which would obey some analogue of Determinacy, or other positive anti-Choice principles (except insofar as the Boolean-valued models of set theory itself are "mathematically natural").
Finally, in type theory, you can sometimes get around the phenomenon discussed above, that postulates tend to break existence properties, by taking negated or double-negated forms of your axioms. The Coquand et al. paper on the subject is very nice. However, by doing this, one gets back precisely to the realm of the "too negative" axioms!

[1]  I will not talk about the views of pluralists here, and what axioms they take or not take. Partly because I'm not one of them, and partly because some pluralists seemingly have a whole lot of time and opportunity to talk about it, so information about their views is easy to come by.
[2]  But this is not a necessity either: e.g. I'm philosophically a more extreme constructivist than most, but all my work uses LEM, because I prefer to focus on academic opportunities over philosophical purity. I wish things would be different, but fat chance of that.
[3] In the study of proof theory and pure logic, one calls this the existence property. There is an unfortunate terminological clash, where in constructive set theories, the "existence property" refers to a subtly different principle (one that, funnily enough, tends to fail for these systems). In set theories, the so-called "numerical existence property" is much closer to what we simply call the existence property in logic. Canonicity in type theory is again closely related.
