Cheesiest Part?
I know that's a bold statement since this movie was chock full of cheese and "icy" puns....but for me the absolute most ludicrous part:
When Batman and Robin are bidding on Poison Ivy's, umm, services after she has put both of them under her spell.
Batman bids a million dollars, topping some rich old dude who bid half a million.
Robin then pipes up with a two million dollar bid.
Batman tells him: you don't have it...and then proceeds to get in a back-and-forth bidding war up to seven million or so.
If all of that weren't silly enough, Batman then breaks out a credit card to a "cha-ching" sound, with a Batman symbol on it that says "Batman" for the name, and "Good Thru" Forever" as the date, with the brand being "GothCard"...and then says "never leave your cave without it".
This silliness trumped almost any of the 1960's Batman cheesiness.
Of course there were many candidates to choose from....a close second might be: Alfred's computer that protected all of the secrets of Batman, the suits, the vehicles, their identities, etc...was protected with the password of "Peg", which was also written on a picture of his ex-wife directly adjacent to the computer screen.
Whose idea was it for the word "Lisp" to have an "S" in it?