Presumably, the converter is already wired into the trailer
electrical system to provide some charging to the
battery when plugged into shore power. Keep in mind that a 1990's converter will likely not have a very good
battery charger in it, so don't try to maintain the battery that way in long-term storage.
For tow vehicle recharging, you will need to have a wire (with circuit breaker) attached to the TV battery and then run all the way back to the trailer connector. On the trailer side of the connector it will have to be connected to the 12VDC system.
If you have a 12VDC mode on your
fridge, it would be wise to install a relay or isolator that won't let 12VDC from the tow vehicle go to the trailer unless the ignition is switched on -- Otherwise, if the
fridge is in 12VDC mode, it can quickly run down both the egg and tv batteries.
Lots of info in the archives about all this.