How's the renovation going?
Free advice from an old electrical engineer who has some experience with the Streamline trailers. Before replacing anything, install a battery and see what works.
Your question on grounding: Those trailers (and more modern ones like my 1999 Airstream) use the frame as the
12V ground (battery negative), same as automotive systems. I could be wrong here, but what is labeled as "transformer" in the system is possibly a battery charger/eliminator which, in the
1964 Prince I was using, didn't actually do a good job of charging batteries. That was partly because the DC lights were something like 40W each (about 4 amps at 12V). I had to use the 12VDC to run the recirculating toilet on the Prince, and I had to make my own charging circuit to keep the battery up, even while the unit was plugged into shore power. It's likely that it really is a transformer, stepping the 120VAC down to 12VAC for the lights in the trailer while plugged into shore power. That would imply that there's a switch to select either battery or transformer power in the trailer. You should be able to find the schematics on the web somewhere these days.
From an engineering point of view, it would be best to connect your battery charger directly to the battery, so it can sense the voltage AT the battery, thereby maintaining a more accurate state of charge - and avoiding overcharging due to voltage drops in the wiring between. That said, there are chargers with "remote sense" lines that can be connected at the battery to improve their performance. IMHO, that's the best way to go, but not the cheapest.
My son insists that the best way to go is to convert to LiFePo batteries. They are lighter, can withstand deeper discharge than Lead/acid cells and last much longer. They are NOT cheaper to install. I have not made the switch yet on my trailer, but will do so the next time I have to buy batteries ($260 for a pair of 220 AH 6V golf cart batteries). LiFePo don't handle the cold as well as lead/acid.
As as example, my 1999 Airstream has a battery charger system installed at the front of the trailer, connected to the battery via the large cables that come to the 12VDC fuse block. (And fused at 50 amps.) This keeps the battery at 13.6V while parked on ground power. Since I replaced all my
12V lamps with LED bulbs, I can light the entire trailer on about 3 amps, instead of 30. Lights can stay on all night, I guess.
12V is also what runs the central heat system in the Airstream, plus it's required for the water heater and the fridge. (I don't know what's in your Streamline, but in the Prince, it was a wall unit run only on propane.)
OK, I'll leave you with those thoughts. Hope I helped. I'll check back later.