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It seems like a very constrained version of the "count integer points inside a polytope" problem. For fixed dimension there are polynomial solutions for it. See: A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension Is Fixed [AI Barvinok · 1994].
I don't know for sure, but possibly you can maybe simplify this paper a bit and just implement that. Still that seems really hard. I don't know if there's any easier ideas.
Thx, I’ll give the paper a read. appreciate your help here!