Now here is the weird and confusing part. If the above is a valid proof, then H will eventually find it. It searches all proofs, remember?

Fortunately, H will never find your argument because it is not a correct proof. You rely on hidden assumptions of the following form (given informally and symbolically):

 If φ is provable, then φ holds.
Provable(#φ) → φ

where #φ denotes the Gödel number of the proposition φ.

Statements of these form are generally not provable. This phenomenon is known as Löb's theorem - featured in Main back in 2008.

You use these invalid assumptions to eliminate the first two options from Either H returns true, or false, or loops forever. For example, if H returns true, then you can infer that "FF halts on input FF" is provable, but that does not contradict FF does not halt on input FF.