What date would that might have been?He didn't stumble over it, it was very fluent production, so he either meant to say it or didn't notice what he'd said. But we seem to have a double modal construction here, something which is not found in Standard English and is attested but not common in certain dialects.
The modals are would and might, and if we put the sentence into a declarative form, you can see what the issue is:
That would might have been what date.Either modal on its own is fine, but both together is not permitted in Standard English. As this is not part of my dialect I can't be sure that this particular combination is allowed in any dialect, but certainly two modal verbs can co-occur in many people's speech.
Not, however, in the antiques expert's speech, I'll bet. I would put money on this being a performance error, which went unnoticed because the fronting of the first modal would means that it's not adjacent to the second modal might. I would guess that he started out asking what date it would have been, and switched halfway through to asking what date it might have been, and the two met in the middle in a sticky mess. Perhaps the much higher frequency of would-questions than might-questions had some influence too (frequency estimation not based on any data or actual facts at all).
This kind of thing makes it so much harder to do dialect syntax through data collection. You might only have a few instances of double modal questions in hours of data, if you're working from interviews, and if a couple of them might be performance errors, how can you be sure of anything? This is why dialect syntacticians have to be cunning as a fox who's just been appointed Professor of Cunning at Oxford, and devise data collection methods that they think will cause people to use more double modals, but without telling them that they want them to use double modals. And getting people to say something in a certain way is really bloody hard. Normal people seem to have this quaint idea that what you say is more important than the way you say it.