How can we trust second-order logic?

Question

Let’s accept for the sake of argument the ontology of set theories without proper classes. Sets (improper classes) are the only things at all. This is perhaps silly, but it will hopefully illustrate a more general point. If we want to regard predicates as things, then they must be sets. Typically this is done by regarding a predicate as the set of all ordered tuples satisfying it. But this would involve proper classes, so we instead need to model the second order logic of set theory as having a universal set. But this will force the membership relation to be interpreted as something other than itself. So why should we trust anything proven using second order logic when its models might not include “real” universe. Things could conceivably be true in all of them without being true in the real universe. I suppose Lowenheim-Skolem might help, but wouldn’t model theory be infected by the same problem, since models would have to be sets? Sorry if this is unclear, I suppose the broader issue is, how do we know that metamathematics doesn’t force us to give up certain models of theories?