If you look at the non-confederate who got nipped by the 14th, the process was he got elected as a representative, and they then held a floor vote that determined he was ineligible due to an espionage act conviction.

Yes, he was already ineligible the election happened before the conviction, but that was the process for unseating him.

I don’t know how that would apply to a POTUS (probably take both houses of congress,).