Generalize Wu formula to integral cohomology classes

I do not think such a cohomology operation can exist, as it would descend to a rational cohomology class in $H^{n+4}(K(\mathbb Z,n);\mathbb Q)$. But for any odd $n$ and also for all even $n >4$, this cohomology group is zero! This in turn gives that $P$ is trivial after rationalizing, so the formula you want could never be true.

The Wu classes are, or at least can be, defined by $Sq^{d-n}(x_n) = u_{d-n}\cup x_n$. The interesting part is not that the Wu classes exist, it is that they are related to Stiefel--Whitney classes, which is not at all a priori. Indeed, this implies that SW classes are homotopy-invariant objects (since $Sq^i$ are), which is not at all god-given. Most characteristic classes are not homotopy-invariant, and so will not participate in a Wu-type formula in terms of cohomology operations.