Let's talk about V, the class of all sets. We can't talk about this in ZFC, of course, so we will use Morse-Kelly set theory instead, which has classes.
My question is, can Morse-Kelly set theory prove (or disprove!) that their exists a model (U, $\in_U$) of set theory (where $U$ is a set, and therefore $(U, \in_U) \in V$) such that (V, $\in$) and (U, $\in_U$) are elementary equivalent as models of set theory?
Yes. MK can define the satisfaction relation for $(V,\in)$ (and more generally for class models), and so it can carry out the usual proof of downward Löwenheim-Skolem for $(V,\in)$ (using global choice to get Skolem functions). This produces a countable elementary submodel $(U,\in_U)\prec (V,\in)$.
Alternatively, if you just want a structure that is elementarily equivalent to $(V,\in)$, since MK can define satisfaction in $(V,\in)$ it can define the theory of $(V,\in)$ and knows that this theory is consistent. The first-order completeness theorem now shows the theory of $(V,\in)$ has a (set) model.