Constant-depth threshold circuit for $\mathrm{PP}$
Yes, Anuj Dawar and Yuguo He claim in their paper titled "Parameterized Complexity Classes uder Logical Reductions" [1] that $\mathrm{SAT}$ is $\mathrm{NP}$-complete, even under quantifier-free first-order projections, which are reductions even weaker than $\mathrm{AC}_0$ reductions.

The same result also holds true for $\mathrm{PP}$ with $\mathrm{MAJSAT}$.

First-order projection can be done in constant depth.

All that is left to be shown is proving that $\mathrm{MAJSAT}$ can be decided in constant depth. Indeed, our threshold circuit for this language has an overall $\mathrm{MAJ}$ gate as the output gate that receives the result from exponentially many result of every possible assignment to the variables of the given 3-$\mathrm{SAT}$ formula. And for each variable assignment, we can, in constant depth, evaluate the formula with this assignment.

[1] Anuj Dawar, Yuguo He, Parameterized Complexity Classes uder Logical Reductions, https://www.cl.cam.ac.uk/~ad260/papers/mfcs09.pdf