you have to check things out systematically.
1. do you have propane it the tanks?
yes then
does the propane stove light?
yes good.
then you have pressurized propane in the system and I would start looking at the furnace.
even though it comes on and runs.. if the battery power is too low.. it will not blow hard enough to make the sail switch in the heater make contact and turn on the propane.
Do you hear clicking after the blower runs for a min or two? if so that means that the igniter is working and trying to light the furnace and the sail switch is making contact..
I would try to re-seat the connectors on the furnace maybe one of them is not making a good connection and preventing the propane valve from opening.
remember that portable heaters do not heat the basement.. and you will have frozen pipes. I would winterize first before you get into additional trouble.
I can't see how it would be the thermostat.. as it is signaling the furnace to come on.. so it is working.