Just to clarify, are you expecting that there is some normalising factor $g(x)$ such that as $x \downarrow 0$, the distribution of the random variable $g(x)\int_0^{1-x} (B(t+x)-B(t))^2 \, dt$ converges to a non-trivial probability distribution on $[0,\infty)$? ["trivial" would just be convergence to 0.] And are you expecting that this limiting distribution can be expressed in some "explicit" manner by some formula potentially involving other well-established distributions? If so, are you able to provide some explanation or intuition behind why you're expecting these things?

In your last sentence (before "The claim follows"), are you not assuming that $\mu$ is ergodic, as opposed to just invariant? [Incidentally, I think the dominated convergence theorem should imply that the whole sequence $\left(\frac{1}{N} \sum_{i=0}^{N-1} f^i_\ast\nu\right)_{N \geq 1}$ - without having to take a subsequence - will converge weakly to $\mu$ as $N \to \infty$, and so $\mathbb{P}$ is just equal to $\mu$.]

Thanks. I'm a bit confused - I think I've just realised that any example of the Bowen-Mañé phenomenon (physical measures that are not ergodic) will immediately be a counterexample to my claim, and yet you have proved that the claim is true.

Sorry, I'm asking questions that I should probably find the answer to by just learning some basic formal logic; I see that what I was suggesting in my above two comments is rubbish, as reflected in mathoverflow.net/q/331897/15570 and mathoverflow.net/a/40826/15570.

I do realise that the way I had formulated my second concluding thought seemed to imply that the people who answered the linked question about Kronecker's acceptance of a proof of Goodstein's Theorem had incomplete understanding of the issue - I don't know if that is connected with the downvotes, but I've now re-written that bit.