How can I install an interior cat door to the basement and not have it kill me? (from increased radon)
If you have a Radon Mitigation system, then you should have no worries. A proper mitigation system puts a negative pressure on the underside of the basement slab, and vents it out the roof. The idea is to remove all radon from the basement itself.

So, in theory, a cat door should have no bearing on the amount of radon in the first floor.

I'd, of course, do a radon test in the basement to be sure, but you should be fine.