The gas burner in an RV refer takes so little gas it is really not of a concern. You will use more gas making a pot of coffee.
As for solar? Solar is useless unless you have enough storage capacity. So first things first, where are you going to place two or three more batteries?
Your next big issue when boon docking is water, both fresh and waste. Most places do not allow waste water being dumped on the ground, so forget that thought.
Storage tank sizes are really going to a problem with that trailer I would think for than a night or two boon docking.