I worked in automotive fuel injection design until I retired 10 years ago. At times, additional 120VAC test equipment was required for gathering data. This was mounted in the trunk along with a second battery and an inverter. Of course the battery was wired in parallel using #12 or #10 wire. Frequently, by the end of the day, the second battery was dead, even though it had been charged overnight.
Discussion with the people who designed the vehicle charging system basically said it was a losing battle. Modern charging systems are designed to recharge the vehicle battery after starting or after a very brief demand for electric power
(electric power assist steering). There will be adequate power to run "continuous" loads
(lighting) but there is just not enough additional power available to charge a second battery, especially one that has a load connected. Also, most light vehicle charging systems are controlled by the PCM which is optimized to only charge one battery.
To confirm this, connect a DMM up to a 12V source on the towed vehicle (12V power point). Dry the vehicle and observer the voltage. It will be >13.2V. Now connect it to your towing vehicle and start towing. I'll bet it is <13.2V.
As silly as this sounds, IMHO, the BEST solution is of these
- Run an AC inverter in the towing vehicle through a normal AC extension cord to a battery maintainer in the towed vehicle
- Use a DC-DC step up power supply to insure that the battery in the towed vehicles is getting a constant 13.2-14.4VDC
In both case, you need to carefully monitor the liquid level in the towed vehicle battery. Top off when necessary and if use method #2 adjust voltage as required.