Wire them both up so that your cheap thermostat provides hard upper and lower limits.
Your HVAC doesn't care where it gets a 24V signal. An open relay won't mind being energized from the "wrong" direction. AFAIK, thermostats won't tattle.
They got back to me and wanted to do a phone call to unenroll me. I told them they could do it without speaking to me since I don't have any power in the process, so we got to skip the phone call. I'm officially out of rush hour.