Yes you can do them both. Your problem right now is how the radiant floor is controlled that causes the temperature to overshoot the set point. Your situation requires a thermostat with PID (Proportional, Integral, derivative) capabilities that self tune to your set points and learns to adjust the run time of your radiant system until it finds the perfect setting. These aren't cheap and aren't the Nest or ecobee types. They're made for radiant floors because of the big lag causing temperatures to overshoot and contain microprocessors and some also sense outside temperatures to better control the system. Here is a link with a good selection to pick from but picking the cheapest isn't the best, you get what you pay for.

radiant floor thermostats.

If the radiant floor heating system is hydronic you can also adjust the water temperature entering the loops to reduce to overshoot with the mixing valve that mixes hot supply water with return water from the floor. This will help some but won't be nearly as effective as a proper thermostat.

I almost forgot, No you can't have 80 degree floors but a cool air temperature in the basement without removing the heat in the air, it is thermodynamically impossible without modifications to the basement. Without opening a window you would need an Air conditioner to remove the heat to the outside (waste) or transfer the heat in the air to the upstairs. A few ceiling vents to the upstairs with fans could work, or a entire duct system for a heat exchanger would work. Either way the room temperature will eventually equal the floor temperature if there is no heat loss in the room. Balancing the heat gain to the heat loss is the only way, and this is what the right thermostat will do but you many not have 80 degree floors.

Does your heat system have the sensor in the floor or is it an ambient (air ) temperature sensor? When we installed our basement floor heat we used a slab temp sensor, so the cycling of the heat is determined by the slab temperature. We usually keep it in the upper 70's so that the floor feels relatively warm on your feet, but with that temp setting it doesn't really heat up the room very much.

Bruce

Bruce, is your slab the only heat source? I have radiant heat in both my basement slab and also my first floor; they are my only heat source. Each floor is divided into 4 zones and only for the MBr do we want the air temperature cool, so we just turn that thermostat down. No AC, because we are Up North!

My home has slabs as only heat source, and we like it alot, we always had issues with the floor being extremely cold, but it works really well with heating on specific zones that auto-changes with sensors.

Just to be clear, the slab in my basement bathroom is heated by an electric radiant heat grid under the tile, and the sensor is in the floor. It is the primary heat source in the room, but there is also a heat/vent/van unit in the ceiling to warm up the room more when someone wants to take a shower.

My basement temp will go no lower than about 65 degrees as long as the heat is on for the 1st and 2nd floor, so I don't need a lot of additional heat to make the basement comfortable. I installed the in-floor radiant in the bathroom only because folks like to be warm when they use the shower.

Bruce