Inverter use while boon docking??

Recently while boon docking we arrived at the remote site with a dead TM battery.  Only water we had, we had just poured into our holding tank. Only toilet besides the woods was our TM. Our tent camping friends had a 700 watt ( 1400 peak) power inverter for "car,truck, boat, rv". In order to use our water pump and toilet, we attached the inverter to our tow vehicle battery, ran an extension cord to the TM AC cord, and turned on the inverter when we needed to supply power to the TM. Didn't use lights, used propane for refrigerator and cooking.

Was this an okay arrangement for the TM equipment? Could we have run the furnace fan which Suburban says needs 2.8 amps? (2.8 amps X 110 volts = 308 watts) This particular inverter has a low battery indicator so one is alerted if the tow vehicle battery is being drawn too low.

I've got a lot to learn about RV electricity and would appreciate input from more experienced TM owners.

Thanks,