I don't know how old your Mac was, but I think system info does record battery details. If not a serial, it definitely reads the cycle count, so it may have been possible to cross reference that if you knew the cycle count previously, but of course, I don't blame you for not making backups of all that information and cross referencing it, you should never need to do that in the first place.

That's odd. That's really dumb for those third-party technicians to take that, as (aside from the damage to their reputation and simply not being a good person), it would probably be a degraded battery anyway. Being constantly plugged in is very bad for a battery.

Sounds like that shop may have gone out of business for a reason.