I think this might be a well known result, but I did a quick research and couldn't find a straightforward answer.
So, is Predicate Logic a Regular Language? And if that's not the case is there a non-trivial subset of it that could be a Regular Language? (Like Propositional Logic only with binary operators?).
No, even propositional logic with a single binary operator is not a regular language.
In the usual notation, where parentheses are used to avoid ambiguity, if the set $F$ of well-formed formulas were regular, then the same would be true of its intersection with the set of expressions in which all left parentheses precede all right parentheses (because this "left precedes right" set is clearly regular and the intersection of two regular languages is regular), and the same would be true for the language obtained from this intersection by deleting all symbols except parentheses. But that language is just $\{(^n\ )^n:n\in\mathbb N\}$, which is known not to be regular.
If you use a parenthesis-free notation, like Polish notation, then the well-formed formulas are again not a regular language, by a similar argument using the fact that, for example, $\land^m\ p^n$ (where $\land$ is your binary operator and $p$ is a propositional variable) is well-formed if and only if $n=m+1$.