Are there any sets of axioms that have an efficient algorithm for all provable statements?

While the term "nontrivial" is still a bit unclear, there's a good sense in which a positive answer to your question is extremely unlikely:

Suppose $T$ does not prove "There is exactly one element in the universe." Then we can in polynomial time reduce the set of unsatisfiable propositional sentences to the set of $T$-theorems. In particular, if $mathsf{Pnot=NP}$ this gives a strong negative answer to your question.

To do this, we argue as follows. Given a $mathsf{SAT}$ instance $theta$ with propositional atoms $a_1,...,a_n$, consider the first-order sentence $$theta':equivexists x_1,y_1,...,x_n,y_n[varphi(x_1,y_1,...,x_n,y_n)]$$ where $varphi$ is the first-order formula gotten from $theta$ by replacing each $a_i$ with the formula $x_i=y_i$. Assuming $T$ is consistent with the existence of two distinct elements, we have that $theta$ is satisfiable in the propositional sense iff $Tcup{theta'}$ is consistent. And the construction $thetamapstotheta'$ is sufficiently simple for this reduction to be efficient.

So basically, unless $T$ is extremely silly we can always embed $mathsf{coSAT}$ into the $T$-$mathsf{THEOREM}$.

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EDIT: And in fact the situation is much worse - as Tom Baker observed, we can in fact get all of $mathsf{PSPACE}$ (the point being that as soon as $T$ has a model with more than one element, we can efficiently reduce the theory of the two-element pure set to the set of $T$-theorems).