Efficient alternatives to inclusion-exclusion
Unfortunately, if the only reasonable method to solve a problem is via inclusion/exclusion, it's going to take $\Omega(2^n)$ time. The reason for this is simple, there are $2^{n}-1$ sets we must inspect (if we don't, then we wouldn't have to do inclusion/exclusion!) So, there are no significant general improvements on inclusion/exclusion.

Asking for an algorithm to "have polynomial run time in most cases" is a rather vague empirical claim that is unfortunately hard to answer precisely and could have little value. A way to make this precise is a probabilistic algorithm. I'm not aware of any probabilistic variants of inclusion/exclusion, but I think that using another method than inclusion/exclusion would be better to create a probabilistic variant.

However, is inclusion/exclusion worthless? No. Often, inclusion/exclusion is used to obtain an algorithm that takes exponential time, but has only polynomial space usage. (sometimes, exponential space is a bigger problem in practice than exponential time!) For instance, the Hamiltonian cycle problem can be solved in $O^*(2^n)$ time but only polynomial space by inclusion/exclusion.



It could be the case that your specific problem has a more efficient solution, but that solution method then is likely very different from inclusion/exclusion.