you could possibly do a set driven key on the pivot point? i dont know how you would do that though, i dont see any option for objects unless you made a custom attribute then linked it via expressions to the pivot point and then used set driven keys on that custom attribute

i dunno how that would work though...

you could just set another control curve to pivot the object at a different point the same way you did it for the first.

so like... if you did a parent constraint on one control curve then do a parent constraint with another curve then you get two constraints showing up in the channel box for the object that's constrained.
then you can just key those (or use set driven keys or expressions)
probably a messy way to do it though... i dunno... no one seems to talk about good practice anywhere when it comes to rigging...