If we are trying to change P, it seems like we can just talk about changes to P rather than introducing u.

Here is the first thing I would try:

In the 1:0 case you can just condition on x, replacing each P(E) with P(E|x). In the general case you instead condition on the fact that a biased coin came up “heads,” where the bias depends on x. (Of course you can also just make the corresponding update directly.) These are operations that your AI already needs to perform every time it makes an observation, so this doesn’t really complicate the design at all.

This is almost the same as your (2), except you omit the normalizing factor (P(x) + lambda P(not x)). The unnormalized version seems theoretically wacky; it is also problematic, since (as you point out) it introduces significant incentives to distort the value of x.