I suppose it depends on how well you ventilate. If you can give the propane heater air from outside the cap you should be fine. There’s a YouTube on using a similar propane heater while camping in a tent which might be useful — Carbon Monoxide Test using a Propane Heater in a Tent - 16 Hour Test. (Mr. Heater)... He has his tent set up to cross ventilate with lots of fresh air by the heater.

The CO detector with digital readout is an excellent idea — Kidde Battery Operated Carbon Monoxide Alarm with Digital Display. I’d take any CO build-up above zero as a bad sign. Maybe I’m over-cautious.