After a discussion with Russell Miller, I've got an answer to the question. There is in fact an approachable real that is not provably approachable.
Let us adopt PA as the base theory, although the construction can easily adapt to other theories. I will describe a Turing machine $M$ that produces a convergent sequence of rational numbers, but any Turing machine $M'$ that PA proves to produce a convergent sequence of rational numbers converges to a different number than $M$ does.
The idea is to diagonalize against any such proofs that may be discovered. Fix an enumeration of the Turing machines $M_n$. Consider the following Turing machine $M$. Our machine will at stage $k$ produce a rational number $r_k$ by giving finitely many trinary digits, but using only the digits 0 and 1 and not 2, to avoid non-unique readibility issues. As the construction proceeds, we systematically enumerate all possible proofs from the theory. We may find at some stage $k$ that there is a proof that Turing machine $M_n$ produces a convergent sequence of rational numbers. In this case, we run $M_n$ for $k$ steps, getting the current rational $q_{n,k}$ approximation to the real to which $M_n$ is converging. In this case, if $r_k$ is different from $q_{n,k}$ by digit $n$, then we use $r_{k+1}=r_k$; otherwise, we produce $r_{k+1}$ by flipping the $n$-th digit of $r_k$ from $0$ to $1$ or vice versa to ensure that $r_{k+1}$ is different from $q_{n,k}$. (Note, we are flipping the $n$-th digit, not the $k$-th digit, so each machine $M_n$ is tied to the $n$-th digit of our limit real.) And simply proceed with this plan, considering the new proofs as they appear.
Note that each machine $M_n$ that provably produces a convergent sequence will be considered infinitely many times, for increasingly large $k$, since there are many proofs that it does so. So the relevant $k$ for $M_n$ will become arbitrarily large. Since $M_n$ was proved to produce a convergent sequence, it follows that the values of $q_{n,k}$ really do converge, and thus eventually stabilize in their first $n$ bits (if those bits are all $0$s and $1$s). Thus, we will flip the $n$-th bit of our rational $r_k$ at most finitely many times. It follows that our sequence $r_k$ is a convergent sequence of rational numbers.
But also, it follows that whenever there is a proof that some machine $M_n$ produces a convergent sequence of rational numbers, then the limit of our real will differ from the limit of that machine by digit $n$.
Thus, we have an approachable real that is not provably approachable, as desired.
Let me remark on the confusing subtle point here about in which theory we have proved our machine to produce a convergent sequence of rationals. The answer is that we have done so in any theory that knows that any statement that is provable in the base theory is in fact true, since we needed to know that when we found a proof that $M_n$ produces a converging sequence of rationals, that those rationals really did converge. For example, ZFC has this relation to PA, since ZFC proves that if $\varphi$ is provable in PA, then $\varphi$ is true. But more generally, for any $\omega$-consistent theory $T$, we may extend to a stronger theory $T^+$ that knows this implication.