I have electric heat in my basement, 240v but 120v can be done similarly... I make a relay box like in this thread (
https://www.garagejournal.com/forum/threads/can-i-control-a-contactor-with-a-thermostat.388491/) and connected it to a Honeywell wifi thermostat (
https://www.amazon.com/gp/product/B00Y6M2OUC/?tag=atomicindus08-20). Relatively cheap and easy to make, and you can use whatever thermostat you want.
I was unsure of the amp draw of my basement heaters so built my own as per that post to have higher amp capacity. If I knew the load for sure and it was smaller, I'd be inclined to use one of these read to use relay+transformer boxes with a wifi thermostat.
Aube 240v RC840T-240:
https://www.amazon.com/dp/B0037MXM1C/?tag=atomicindus08-20
Aube 120v RC840T-120:
https://www.amazon.com/dp/B00D5YLY2G/?tag=atomicindus08-20
Forget about wild 5* temp swings. Any modern thermostat will work fine and maintain temp to what you set. The Honeywell RTH6580WF I use in my basement with electric and my garage with a gas unit heater goes down to 40*. Initially I set my garage thermostat to 50* as that's the lowest recommended by Sterling for my non-condensing heater to avoid condensation and rusting out the heat exhanger prematurely. I later set it ot maintain 60*, returning to 60* every 6 hours, as I can raise temp to my desired 72* within in hour and it hardly runs any more at 60* than 50*. With electric you could set it to 40* and be fine, but if you want a reasonable working temp of 60*+ it'll take forever with 120v heat. If electric is the only option I'd go with 240v.