Should I buy or hack a 240v remote controlled thermostat?
240V thermostats (of 10A capacity or usually greater) are easily bought off the shelf - they are commonly used to control electric baseboard heat from a wall-mount box. You could mount one in a freestanding box if you want this "portable" as well, or you could wall-mount one and have it control a (labeled) outlet that the heater would plug into.

While you could also use any thermostat and a relay with a suitable coil voltage for the thermostat, that's just more parts and complexity for a problem that there's a one-peice solution to, commonly available.