Hello and welcome.
this is my personal experience for the last few years and as recent as a week ago, as I just got back on Monday from 11 days away again.
We have a 2008 tent trailer (prior one was 1991) and we have taken our birds with us 3-4 times every year and as late as October. Our furnace comes on and off as needed, and like yours is vented out and we have never had any issues. We always leave the roof vent or window flap slightly open for fresh air flow though. The trailer is always comfortable even though we have gone down to 10 Celsius and through rainy days as well.
i feel comfortable taking my previous babies and now Bailey with us.
if you have a new trailer though, you might want to burn in the furnace for for a few hours daily before you go.
my two cents worth, though I know it’s not for everyone.