It's substantially harder than linear programming: it's equivalent to SAT, whereas linear programming is merely polynomial-time (and in practice weakly polynomial-time with current algorithms).

I normally use Simplex method which is fast and not polynomial in the worst case though

You can always just run a portfolio of Simplex/Barrier/PDLP and just grab whichever returns something first. The latter two are usually slower but polynomial time, so you always win.

Can't do that with SAT or ILP.

Simplex usually runs fast, and Barrier or PDLP help with LP. But for SAT or ILP, there’s no quick way. You can’t just try a bunch and pick the fastest. Those problems are just tougher.