Integer programming to MAX-SAT translation
The decision versions of both MaxSAT and integer programming are in fact NP-complete, so there is polynomial reduction from integer programming to MaxSAT.

In the context of solvers, modern MaxSAT solvers support "weighted partial MaxSAT"-encodings (weighted clauses with possibly infinite weights), so you can add any SAT encoded hard constraints by encoding them as clauses with infinite weight.

For comparing MaxSAT and IP solvers, I would strongly suggest looking at modern literature on "implicit hitting set" algorithms for MaxSAT. The solvers using this algorithm use both IP solvers and SAT solvers as subroutines for solving MaxSAT. It might even be that if you use some direct translation from IP to MaxSAT and give the resulting instance for these solvers, the solver will under the hood just directly solve the original IP instance with the IP solver. Also, there is a lot of recent research comparing IP and MaxSAT solvers for solving specific problems (just search "integer programming maxsat" in Google Scholar).