My radon manometer has a red reading, what should I do

I wouldn't even consider one or the other... both.

An outdoor sump pump in our could climate (I'm in Minnesota) is a real pain in the neck. A good friend of mine has one in his old house and he has to bring it in every winter and put it back out when the snow starts melting, which often means a few hours digging to find the top of the sump pit, then every couple days checking and adjusting the position of the pump as the solid ice on the bottom of the pit melts. Worse was the year it melted, then re-froze with the pump in it... I recall he had a number of very unpleasant things to say about it the next day at work. ;)

Depending on your levels of radon, the mitigation could be pretty simple. Several of my co-workers have a small bathroom exhaust fan in the utility room where there sump is located that ducts out of the house. It's on a timer that runs it for X minutes every Y hours, depending on what the levels were in their house. Most of us agreed that that was better than having to deal with flooded basements every few years.

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.