You would need a second trailer to haul enough batteries to run a space heater for any significant time. Adjust your expectations to running the furnace with the battery power and propane.
2000 watt inverter could run any home coffee maker and your tvs. You can get the inverter with a built-in transfer switch and battery charging when plugged in. I always recommend sine wave.
The solar controller is your smart charger. Review the available charge programming before you buy. Not sure why you need blue tooth unless you need one more distraction.
If you find a kit with all the right parts I am sure that would make installation easier. I matched my own components.
Enjoy the journey.