Heating a trailer while under tow

I don't know any reason you can't run the furnace while traveling. Many people, including myself do it. There's nothing in my owners manual that says not too.
As for a 12v heater, there's no such thing as a 12v heater that would make any difference and also "not draw so much power as to deplete the batteries". The best you might get thru a factory charge wire is about 15A. That's only 180 watts.

When we traveled with all the house plants we always had a catalytic heater running on the road.Always traveled with the catalytic heater on in our motor homes and campers.

Put the plants in the bathtub with door fixed open for the heat.

As suggested already, I would probably just run the furnace and keep an eye on things. Not sure if it's really recommended, but I've heard of others doing that w/o problems.

I recently bought a second camera for my non traditional backup camera system and have one to monitor the main interior space in the TT while on the road.

Larry

Check to make sure your tow vehicle is supplying 12v down the charge wire to the trailer. Preheat the trailer on shore power with the furnace running for 12 to 24 hours depending on the weather. Get it good and warm inside. Make sure the battery is charging from shore power like it should.

Load up and go, leave with full lpg bottles and leave the furnace running, plug in to some shore power during overnight stop overs if possible. 15 amp is lots to run the furnace.

The furnace uses fuel much faster during winter driving than summer vacation. Our truck camper will use close to 20lbs per 24 hours travelling in -10C to 0C conditions. It is 9ft with 16,000btu furnace. Depending on weather conditions the bottle that is supplying the furnace will have water vapour condensing on it or freezing on it. This can be an accurate "fuel guage" if liquid but I prefer the shake the bottle method in cold weather.

Safe travels.

What about a 12 volt space heater? Something big enough to keep the inside warmer than freezing, but not so large as to risk running down your house battery faster than your vehicle could keep it charged. No open flame, and you could set it pointing at the plants. Just make sure to secure it while you're going own the road, obviously.

No concern there, mine doesn't have any.

I aways ran the furnace in my previous TT while on the road. Not a problem. As long as any slides don't cover furnace vents.

some of the house plants are over 4 feet tall.

If the houseplants aren't too big, you could put them in an ice chest with a couple bottles of hot water.

I have moved my TT many times in the winter months. I am fulltiming in Calgary in it right now. I run all over the place for my work and usually drag it with me.
Although I do hesitate about running the furnace while driving, I have done it a number of times. When the temperature is down around 0F, I can't run the risk of a total freezup while driving.
I have never had a problem doing this but have to admit I am a little apprehensive about it.

Worst thing that ever happened was a full pound of butter slid off my counter top and landed neatly on its side on a heat register. Quite a mess when it all melted and ran into my heat duct....
I still get this weird urge for popcorn sometimes when I run the furnace for a long time....

Not endorsing running with the furnace on but I have done it many times without any incident aside from the butter trip.