This is in a language with identity?

Suppose M is a model of T and Q is a model of T.

Let UM and UQ be the respective universes.

M maps P to some subset, call it M*, of UM; and N maps P to some subset, call it Q* of UQ.

If UM is finite, then, by completeness, there is a theorem of T that specifes the exact finite cardinality, call it n, of M*, which, by completeness, is the exact finite cardinality of Q*.

Now, from the given bijection between UM and UQ, and by induction on n, show that there is an isomorphism between M and Q.

Otherwise, UM is denumerable. Here, if I'm not mistaken, from the given bijection between UM and UQ, you can still construct an isomorphism between M and Q, by using strong recursion on UM (i.e., you might as well consider UM to be the set of natural numbers). You might have to do some "gluing" of functions together, but I think it will work.