It's clear that, as a first pass, Hashcash is in FNP.
For instance, say that R(x,y) is a binary relation, where x is a "maximum" target hash, and y is an input string (working block + nonce), R(x,y) is true iff hash(y) < x. Since SHA256 can be computed in polynomial time, proposed solutions can be checked in polynomial time.
It's also clear that, furthermore, it's in TFNP, since for any x (hash target number), a string y that hashes to less than it is guaranteed to exist.
It's also clear that, furthermore, an infinite number of such y exist for any x - there are an infinite number of strings, but a finite number of hashes, so a solution to this problem is guaranteed via the pigeonhole principle.
Can we use this, or any other similar insights, to identify hashcash as a member of a complexity class finer than TFNP? Would it be in PPP? PPA? PPAD? Something else?