Keep it separate.
I would use a low voltage thermostat because it's easier to run LV wiring around than romex. Use a low voltage stat like a White Rodgers heat only 1C20R (looks like a Suburban one). Use a small 120/24 volt transformer, 20VA. Use a heating relay like a Honeywell R841 with or without integral transf. You can get a heating relay that is instant off like an AUBE RC840 (not cheap tho.)
I would suggest installing a line voltage switch so you can turn the power off when not there, parked at home or you want to plug in a heavy load. The Honeywell R841 takes a minute or longer to turn off the heating load when the call for heat is removed. Also, a low voltage stat. has a heat anticipator that delays the off - you can turn it to the lowest setting but it won't be instant off.
You could also install a line voltage stat. with an off indication if you can get the wiring to the stat. location.
Not sure where your heater will go and where the existing t'stat is, but if you put a new LV stat beside the existing, there's very likely a hole in the top plate of the wall that you can fish a new wire through. I've done that a few times now.
Rather than using that Cadet heater, can I suggest another one? Look at the King PAW "pic-a-watt" series heater
King PAW specs.. You can set the wattage in 250 watt increments from 250 to 1500 watts. Also, because it has a tangential blower, it is super quiet. I am installing this one in our TT. It's so quiet, you have to put an ear up to it to check that it is on. I think you will find that the Cadet heater (or similar ones) will be rather noisy because it has a propeller type fan. I installed a similar one in our first TT and it was really noisy and I'd never use another one like again in a TT. Most wall heaters have a propeller fan.
Assuming that you have the correct clearances from wall and other surfaces, one thing you may want to think about is what happens when head back home. If you plug in at home, is there enough clearance when slide(s) is in and what if you unintentionally leave the heater running? You could install a relay to latch the power on and will only re-energize the power by hitting a momentary contact switch - easy to do. I'm using a Finder #66.82.8.120.0000 relay to do this.
Finder power relay
