If you've got 120v, your converter will supply power to all of your trailer's 12v items (lights, water pump, furnace, propane detector, your fridge's brain, a slide out, etc).
And, yes, you'll need something small to power the breakaway system - a riding mower battery would be fine.
Without a battery as a buffer, I don't know if your fridge will be able to reliably draw enough power from your vehicle en route. Probably.. if you've got 8 gauge wire (or better) directly from your vehicle's battery to your bargman connector for aux power.
But, honestly, two batteries and two propane tanks will weigh about 180 pounds. Net: 160 pounds drop by the time you add a small battery. I doubt that you'd see any gasoline savings by dropping that weight. It would certainly be no more than a 1% increase; $1.50 savings on $150 worth of gas for a round trip. I just don't think its worth the effort...