Low for EXP and NEXP
$\mathsf{BPP=NEXP}$ is consistent with our current knowledge, thus, since $\mathsf{EXP, NEXP}$ are not low for themselves, it is not known whether $\mathsf{BPP}$ is low for them. A positive answer would imply $\mathsf{BPP\neq NEXP}$, and a negative answer would imply $\mathsf{P\neq BPP}$ (and in turn, that $\mathsf{P\neq NP}$), so currently, if you're looking for unconditional results, you will have no luck with $\mathsf{BPP}$.

Since even a (possibly weak) class as $\mathsf{BPP}$ is not known to be low for $\mathsf{EXP}$, you will need to look between $\mathsf{P}$ and $\mathsf{BPP}$ (which does not give you much wiggle room), or for classes who are currently incomparable with $\mathsf{BPP}$ (such as $\mathsf{QP}$ as you yourself noted, I doubt you can find a significantly stronger low class for EXP).