I don't think there's much you can do with such a character. Unless you place her in a coma and when she comes out there's a personality change. Or she could always fall down some steps and bash her head. Or perhaps get alcoholic poisoning....
Or Stan could always crush her under his weight.... Come to think of it now, the possibilities are endless
Yes, the possibilities are endless...that is IF you have great writers, which is the key to any successful sitcom. There are a gazillion of situations they could put the "Karen" character in to further develop the character, not to mention bring fresh new characters in for her to play off of. Add to that the occasional cameo appearance by "Will & Grace", and the ratings could soar
No I don't think it would work.
What made the character fun for some people was the "shrill harpy drunken drug addledness" as Blondizgd said.
This is too one dimensional. If what made Karen, Karen is changed too much then the character becomes unrecognizable and would no longer be the character that people loved. If she remained as is - then there's no point to the spin off - she has already been done- on W&G.
Think of Christine Baranski’s character from Cybill, which was a very similar sort of character to Karen. Wasn’t there a spin off of Christine’s character?