I have a 92 Buick park Ave, (not the Ultra) ... SAME ISSUE!!!
I am a (expired) licensed Master Mechanic ... NEVER seen anything like it. New battery, completely charged numerous times, dead every morning. Alternator putting out a good 14.5V, all battery connections clean and tight. No starter issues.
A real key surfaced the last time I charged it: Flat dead battery. No keys in vehicle. I hooked up the charging cables, like it was possessed, all manners of clicks and rrrrr's, radiator, A/C, and cowl fan (interior) cycled on and off (and if I can remember clearly from my shocked brain, not in sync). No issues when being operated.
So, here is my highly technical answer (I reserve the right to deny this at any time) ... it is possessed.