Terminology: I'll denote this theory by "Mereological Logicism", because its primitive are Part-hood and, Predication, which are the primitives of Mereology and Logic. However those are here extending first order logic, so it is a first order logic theory about Mereological Logicism.
EXPOSITION: To first order logic with identity, add the binary relation $P$ standing for "is a part of", and also add the primitive binary relation symbol "$predicates$".
Axioms: ID axioms plus Atomic Extensional Mereology "AEM" axioms, plus the following:
Atom Predicates: $\forall Q [atom(Q) \leftrightarrow \forall x (Q \ predicates \ x \to atom(x))]$
A predicate is an atom if and only if all of what it predicates are atoms.
Predicate Existence: if $\psi$ is a formula in the language of predication, i.e. only the symbols $``Predicates; ="$ are allowed to be used as predicate symbols (i.e., the symbol "$P$" is not allowed), that has all and only symbols $``y,x_1,..,x_n"$ occurring free in it; then:
$\forall x_1,..,x_n [x_1,..,x_n \ are \ atoms \wedge \forall y (\psi \to atom(y)) \to \exists Q \forall y (Q \ predicates \ y \leftrightarrow \psi)]$
, is an axiom.
If a formula $\psi$ in the language of predication, is closed on atoms; then a predicate exists that predicates all and only objects holding of $\psi$
Extensionality: $\forall A \forall B [\forall x (A \ predicates \ x \leftrightarrow B \ predicates \ x) \to A=B] $
Predicates having the same predication are equal.
/Theory definition finished.
Now if we define the membership $``\in"$ as:
$y \in x \iff x \ predicates \ y$,
then I'd think we can get to interpret Ackermann's set theory minus class comprehension, which [I think] in turn would interpret ZFC.
The strong point is that this theory is almost a theory of logic! In some sense signaling revival of the program of Logicism!
So the crucial question: is Ackermann set theory minus class comprehension equi-interpretable with this theory?
Another interesting question is if the axiom schema of Unrestricted Composition Principle, can be added to the above system? which would possibly interpret full Ackermann's set theory.
Note: should unrestriced composition proves inconsistent with the above, then I'd think that the following minor modification of the first axiom, and asserting existence of atom predicates in predicate existence scheme, should work: $$\forall Q [atom(Q) \leftrightarrow \exists A (atom(A) \wedge \forall x (Q(x) \to atom(x) \wedge A \ predicates \ x))]$$
The idea of the later question, is that if true, then it would enable us to speak of extensions of atom predicates as Mereological totalities of atoms that satisfy those predicates. To do that we need to define a new membership relation:
Define "$\epsilon$": $x \ \epsilon \ y \iff x P y \wedge atom(x) $
The membership relation $\epsilon$ need not be confused with the usual set membership relation $\in$. Here $\epsilon$ can be termed as: 'extensional membership' relation.
We have the following theorems about extensional membership:
i. $\forall x \exists y: y \ \epsilon \ x$
ii. $\forall xy [\forall z (z \ \epsilon \ x \leftrightarrow z=y) \to x=y] $
iii. $\forall xy \ (y \ \epsilon \ x \to \forall m (m \ \epsilon \ y \leftrightarrow m=y))$
The unrestricted composition principle would prove that every atom predicate would have an extension corresponding to it, that is:
$\forall Q [atom(Q) \to \exists x \forall y (y \ \epsilon \ x \leftrightarrow Q \ predicates \ y)]$.
Define: $x=eQ \iff \forall y (y \ \epsilon \ x \leftrightarrow Q \ predicates \ y)$.
Now a predicate $Q$ is said to define an extension $x$ if and only if $x=eQ$, likewise $x$ would be said to extend $Q$; i.e., a predicate defines what extends it!
It is important to note that although every $\in$-subset of an atom predicate is an atom predicate, yet it is not true that every part of an extension of an atom predicate has an atom predicate that defines it.
A last point, with the unrestricted composition added, its plausible to weaken Extensionality to be applied only to predicates that have extensions! Formally:
$\forall A \forall B [eA=eB \to A=B]$
This would allow having many "objects" that are not predicates, which is only very natural!
For an informal intuitive motivation of this theory, click here