Put simply, I added a switch that operates the blower in parallel with the control board. I believe you are correct in your statement that it would be a reason not to honor the warranty if I have furnace trouble. In addition there is another problem. The furnace blower not only circulates house air on one side, it circulates combustion air on the other. Since there is no fire it cools the heat exchanger, in turn cooling the air delivered to the ducts to some extent. There is a simple solution, but it is a safety concern.
Your T connection is a good idea, just make sure there is still adequate airflow from the furnace when it does run and the inline fan doesn't fight the furnace and stagnate flow. A completely separate duct that just heated the underneath would be ideal. Especially if it was easy to remove and stash when the winter camping is over.
I'm interested to hear how the heated hose works out.