If Columbo could prove that the cork in the office and the cork at Lilly’s house were from the same brand of champagne, could that connect Ken to the two murders?

It would be a Hugh Creighton-esque case. It wouldn't necessarily prove it, but if Columbo had noticed which brand of champagne he saw at Franklin's house while the host was fawning over the bizarrely-dressed interviewer (and I'm sure he would have noticed), it would be fairly strong circumstantial evidence, at least in Columbo's mind. I don't think it would have been enough to prevent the LaSanka murder (after all, at that time Columbo didn't know about the Franklin-LaSanka connection), it would only make Columbo's suspicions stronger. It may have made Franklin less careless at the LaSanka apartment, knowing Columbo was looking for champagne clues.

Isn’t LaSanka the same last name in the prison where Jack Cassidy is a writing partner and he kills her because she knows he killed his writing partner?

Yes, Beck, that was the other murder to which I was referring. Columbo found a champagne cork in Ms. LaSanka's house after she'd been killed by Franklin. If he'd found a champagne cork in the Franklin/Ferris office (which he should have, as it's fairly certain Franklin did not clean up the place after trashing it), if Franklin was seen leaving his house and heading for the cabin with two bottles of champagne (which he was) and if Columbo could prove the cork in LaSanka's was the same as the cork in the office and from Franklin's house (ala Hugh Creighton), then he'd have a pretty solid circumstantial case vs. Franklin.