Of course the manufacturer is going to advise against EVER towing with the furnace on. Their lawyers would be foolish to advise otherwise. And, some states do have laws against towing with propane on as they are concerned that in the event of an accident, a fractured line could result in a blowtorch.
However, that being said, I have friends that have towed for over 35 years with frige on in the summer, and both furnace and frige on in the winter, with no problems.
Would think this is an activity that you just have to decide at your own risk given your circumstances.
In winter, I would hate to be towing over a mountain pass where I KNEW it would be below freezing for very long, and have no heat in the trailer....can you say frozen lines?...like say from Artesia NM to Alamogordo NM where the mountain is over 9000 ft at Cloudcroft. The road might be clear, but the temp could be well below freezing on the mountain and well above freezing at Artesia and at Alamogordo.