It would be funnier (if you're going for a comedy) if Clark inadvertently (without realizing it) cleared his name, but more difficult to write. I've noticed that difference between the comics and L&C, too.
Lois Lane Fan has written a good story or two about Cat discovering Clark's secret and trying to prove it, especially to a much disbelieving Lois. It looks like he hasn't posted them to the Archives, though.
/nudge, nudge, LLF/
I Saw Superman Kissing Cat Grant and the other one is... Hmmm. I can't find it. Maybe it was by you, Queen of the Capes, about Cat discovering what Bruce does on his off time. Can't recall the title though.