I would like to install electric radiant heating in my bedroom floors. But all thermostats with floor sensors utilize a powerful relay that makes a loud clicking noise every time it comes on and off. Are there any alternatives that would avoid this problem?

Things I have considered so far:

* Install a thermostat in the utility room, which is conveniently located below the bedroom. This will mitigate the clicking noise, but obviously not desirable, as the thermostat is now nowhere near the floor being heated.
* Place a thermostat in the hallway. Don't think this is going to work, as I will not be replacing hallway flooring, and won't be able to get all the wires there.
* Hack a thermostat to remove the relay, and wire a remote module to turn the heat on and off. Least preferred way, as I'll probably brick at least 1 thermostat in the process :) Plus having the remote module is extra cost
* Hack a thermostat to replace mechanical relay with solid state relay. Not gonna work, as just 5 amp load will generate 5-8W of heat, which can't be easily dissipated from a wall box.

Essentially what I need is a signaling thermostat with floor sensor, and a remote "power module" with GFCI that is controlled by the thermostat signal. Or maybe someone invented silent relay thermostat. Thoughts?

Edit: another thought... Convert 240 VAC into DC somewhere with ability to dissipate hear... Run to a thermostat (assuming it'll work on DC), and hack it to replace relay with power MOSFET. That should still allow GFCI function to work, and MOSFET wouldn't have much power dissipation requirement.

EDIT 2: I do not need to control air temperature with this thermostat. Floor heat is for convenience only, not to warm up the room, so it will be set to a minimal setting just to be nice to touch. I have a forced air system for the air (and I expect it’ll work less when floor heat is on).