cooling my Krell monoblocks

You let a cat spontaneously combust. Could you be anymore illustrative?? My word !
As for the fan and clicks ask Krell to tell you what you can do beside ignoring it which you can't. Be very firm and ask them to get an engineer to tell you that you may or may not use the fan . The big McIntosh amps had fans and may still but as the others have said the Chinese Oops I mean U.S. engineers have taken that into account. Just don't use ice and don't let a cat get toasty on the amps that would warm them even moreso.
Another idea is to go back to the 250s if they had enough juice for you. I bet a whole bunch of click loving 250 owners would trade up for 350s.
BTW who or what is Krell is it a derivative of an English root. I know Baleen whales and Whale sharks etc. eat "krill" a tiney shrimplike sea creature. Krell is owned by an Italian American. I think his is name Dan D'Agostino.

Again, I urge caution.
ALL amps without fans as a designed in feature are passively cooled. The idea is that warm air rises and pulls cool air in from the bottom. Running under normal conditions in a 75f house, this should be enough. Stick kitty on top and finish the job by clogging it up with cat-hair and dust bunnies and you've got an overheating recipe.
Extreme use, like averaging about 1/3 power will stress SS to the max. Add a clog or 2 and there you go.
IF a fan is used, I don't think it should blow AT THE TOP of any gear. This will force air backwards down into the unit and restrict natural convection. Still air in the amp will just get hotter.
Many years ago I designed and built a small (4"box fan) that fit snugly on the back of my Carver Cube. It blew OUT, sucking air thru the amplifier. I could and did run it all day long just short of clipping without it cooking.
In general, I would say that pulling air up/out of an amp with internal heatsinks and pulling air over external heatsinks would be best.

I'll contact Krell and find out what options are available and let all of you know the reply. While I'm at it I can ask the D'Agostinos where the word "Krell" came from. That too would be interesting.

Try using fans to draw the heat away. I use Noctua fans - they are Swiss made computer case fans and the quietest around. The other brand would be Scythe, which are also quiet compared to most fans but a little noiser than the Noctua. At $20 a fan, it may be worth it.

While I'm at it I can ask the D'Agostinos where the word "Krell" came from. That too would be interesting.
Puerto (Answers)

Others have reported that the name came from the Star Trek character, Krell, because the owner is a Star Trek fan.

Let us know if you confirm that with him.