Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program techniques (See Complexity of integer quasiconvex polynomial optimization by Sebastian Heinz.
What is the difficulty in generalizing Barvinok type techniques(https://barvinok.sourceforge.io/) to counting integer points in convex integer programming?