I don't think the situation is comparable.

Recall that in the chess case the setup is this: the verifier is given a string of n random bits, asks P(n) interactive questions to the alien, and then says yes or no. If White can force a win in chess the verifier says yes in 2/3s of the cases, while if White cannot force a win the verifier says yes in only 1/3d of the cases.

It's a deep result that you can set up a verification protocol like this for any problem in PSPACE. But it's easy to see that it only works if the problem is in PSPACE: using P(n) space, the alien can simulate all possible question-answer interactions with the verifier, and give the answers that will make the verifier answer yes if there are any at all. So you could never use a protocol like this to prove that you possess a halting oracle, since even without an oracle you can do a finite amount of computation to figure out what answers the verifier will accept.

I would guess this principle holds in general: no matter which particular notion of "proof" or "convincing" you settle on, it will ultimately involve running a finite computation to verify a purported proof. But such a procedure can always be fooled by another (longer) finite computation.

(Edit: corrected error in the description of the protocol)