How much fork to tyre clearance do dedicated cyclo cross bikes need?
It would be very tight, looking at the bike I think 32c with knobbies might not even fit. It also has to do with the height of the tire. By adding the knobbies I doubt you will find a CX tire that will fit under the brakes. 

There are not many options for 30c or smaller CX tires. If you can find some tires that fit (will be tight) it will clump mud, but might not lock your wheels up. It depends on the mud, not all mud is the same, some sloppy/wet mud will not clump.

But extra clearance doesn't mean everything rolls smooth. The mud in a race two weeks ago on my 29er was so thick that both front a rear wheels locked up. 

All that being said, you can find a CX race that has less mud. Great fun and highly recommended. Another option is to race on a mountain bike to see how you like CX racing.