Many converters of that era had 2 types of outputs: filtered and non filtered. The filtered ones went to the electronics: radio, TV booster, etc. The non filtered to everything else. The battery was used as a way of filtering some of the AC ripple from the converter output. I don't know if those outputs would work or not without the battery but it would seem likely they wouldn't.
Also, it's probably not recommended to run the system without the converter since it was dependent on the battery for filtering and to a certain extent voltage regulation.
And, off course, their is the break-away brakes issue which make the battery mandatory for towing the trailer on a public road.
I hope calamus instals a battery in case his trailer ever comes unhitched. Otherwise he'll need a really long cord to the truck for it to stop.