I won't expound on the safety of a Buddy heater, although, know they are safe. Biggest downside is they put moisture in the air and create condensation as a result.
Presuming your camper doesn't have a built in furnace.
Honestly, it's either not that cold if a little $20 cube heater does the trick, or your more resilient to cold weather than you're giving yourself credit for!
If you can deal with the condensation, I'd be more inclined to explain the safety features it DOES have and also that you have (or should have if any other gas appliances) a CO detector in the camper.
If you weren't aware, they have a tip over switch which cuts them off if...tipped over, or even jostled. And also have a low oxygen sensor which cuts it off in the presence of not enough 02.
While they don't have a specific CO sensor, the 02 cutoff achieves the same effect, as 02 replaced by CO in an enclosure will shut it off just the same.
Can confirm it works well, as they're only recommended to operate up to 7kft altitude IIRC. And I could keep mine running, barely, at a little over 8kft but it would randomly shut off (being a combo of density of the air at altitude mostly and likely a slightly higher CO concentration right at the heater itself) in an 8x12 relatively sealed up, uninsulated tin can cargo trailer. But it was around 0deg F and a Big Buddy running full tilt, vs much less heat needed for your scenario.
You can be 100% safety conscious and use a catalytic LP heater indoors. Just have to understand it, if it's worrisome.
And you'll want some ventilation an airflow anyway and won't be worried about sealing up the camper from cold air if you have a heater that works 10x better.
It is 100% easier to deal with than the battery/charge/inverter/solar/etc/etc of trying to power an electric heater in the middle of the woods with limited battery capacity.
Just food for thought.