We work in General extensional Mereology "GEM". Add to it the axiom that every object has an atom that is a part of it, thereby getting "AGEM". Then add a primitive total one place function symbol $\mathcal l $, to signify "the label of",
Labeling: $\forall x \forall y: \mathcal l(x)=\mathcal l(y) \to x=y$
Pairing: $\forall a,b: atom(a) \land atom(b) \to atom (\mathcal l(\{a,b\}))$
where $\{a,b\}$ is the fusion of atoms $a,b$, i.e. its the object that has atoms $a,b$ as parts of, and that doesn't have any other atom as a part of.
Since GEM enable us to construct freely any fusion of atoms, then we can define relations as fusion of atoms that stands as ordered pair atoms (say in the Kuratwoski sense).
This mean that we can finitely axiomatize any first order theory in the language of "labeled AGEM"!
The reason is because we can define the membership relation $\in$ as being an atom in a mereological fusion. That is
Define: $a \in x \iff atom(a) \land a \ P \ x$
We define an "ordered pair" as
Define: $ordered \ pair (p) \iff \exists a,b: p=\mathcal l(\{\mathcal l(a),\mathcal l(\{a,b\})\})$
Now take any first order theory, take all primitives in the language of that theory, all constants would be considered as "atoms", all unary predicates would be considered as "fusions", and all n-ary (for n> 1) predicates would be considered as fusions of ordered pair atoms (n-tuples). Now we write all axioms of the theory with respect to those fusions and the membership relation defined above, so each "$\phi(x)$" would be turn into "$x \in P$, and each $\phi(x_1,..,x_n)$ would turn into $\langle x_1,..,x_n \rangle \in P$. And so all schemas in that theory would be converted to single axioms. Since all $n$-ary functions can be re-stated as $n+1$-ary predicates with appropriate axioms added to ensure functionality, then the above would suffice.
Question: would the above approach work to provide a general framework for conservatively extending any first order theory in a finite manner?