One classic technique is to reformulate the integer variables as a bunch of binary variables. In this case i would use a encoding with powers of 2 and then use a multiplier circuit to express the multiplication. Since multiplicaiton circuits are made out of ANDs and ORs the resulting problem will be MILP.
Bi-linear terms (and also monomial terms which all can be expressed as nested bi-linear terms) can be over approximated using McCormick envelopes. Some MINLP solvers (i am aware that Alpine.jl does that) use this to express products.
I would recommend throwing some MINLP solvers at is first and see how they do. If that proves infeasible see if you have success with a binary linear formulation and see if MILP solvers help you. If all your variables are binary and MILP solvers fail to find feasible solutions try Exact.