You should consider, but not invest now. Wait and test, before treating. Small design changes now, can make implementing a fix later, easier.
Radon is everywhere, to some level. If you have higher levels of radon, then you need to be concerned with the rate that fresh air is exchanged in your home. Houses are much tighter today, so be more aware with a new house.
Many factors affect where radon collects. Do you heat with wood? That moves air around your woods stove. Do you have leaky windows? That exchanges small amounts of air all over the house.
To get a general sense, ask a few neighbors if they have tested for radon. On the other hand, I have rock under my house, while my neighbor has sand, so things can vary quickly. You could call a few local excavators, or well drillers, as they are likely to have a guess. Fill (the dirt and stone under your lawn) gets moved around so much, that you can't rely on historical or nearby testing.
Your local building inspector is probably going to be your best resource. And since you're going to be seeing that person alot anyway, you might as well make contact.
Let's say you have a concrete slab, with stone under it. Whether or not this is a basement doesn't matter. Your builder will put a layer of plastic under that, to keep out the "rising damp", and it may help limit radon. It would be a simple thing to ask for 8 feet or so, of perforated pipe, capped, to be placed under the slab. In the middle, have a 90 with a pipe come up through the concrete into your house. Cap it for now. $20. When you test later, it may be enough to stick a bath fan on it, and pipe the air outside. Cold ground air is simply piped outside, and radon may not collect.
If you are putting in ducts for heating/cooling, ask about how an HRV could be attached. I think this is the ventilation you were thinking of (heat recovery ventilator). The HRV is about $1k, and could be bought later. The ductwork is hard to do later.
In addition to air quality, you may also want to check on radon levels in your water, if you have a well.
Cost reduction is good. Basements are often incorrectly drained, so can be damp and harbor mold. On the other hand, if you put in good drainage, basements are cheap square footage. You might consider reducing closets and rooms and keeping the basement.
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.
Best Answer
Unless you have had your house professionallly tested for radon and got a positive (i.e bad) result, do nothing.
If you do have high levels of radon, it's much simpler to install an exhaust fan system, either in the basement or (more properly) by cutting a hole in the basement floor and dropping a 6-inch pvc tube down into the dirt. That acts as the source for a closed-system exhaust fan that should be ducted to the outside.