We use our stove to heat up the Trillium
all the time. Did the same with our former tent trailer.
Once we are in the trailer for the night we turn the stove on for a couple of minutes. Vent is cracked as is a window or two. The stove gets the trailer nice and toasty and then we turn it off. The key word here is "TURN IT OFF". Same thing in the morning if its cold, turn the stove on for a few to warm up while making the morning coffee.
I personally don't see any difference in having the stove on to cook with, having it on for a FEW MINUTES to provide heat or to have a catalytic heater on. Those portable catalytic heaters don't vent.
I have mild asthma and react to fumes (and dogs:cry). We got a catalytic heater numerous years ago and ended up taking it back. It started me wheezing, yet the stove never does.
Just make sure its only on for a few minutes to take the chill off and to be extra safe, have a carbon monoxide detector.