Without additional technical details it it is difficult to determine the cause of the failure you noted.
It may well be low voltage under load but that does not necessarily mean the problem is battery or power source related. If everything is working on the tow vehicle it is likely your source power is OK. The problem could be a poor connection or some other resistance component in the brake circuit. After first checking all connections including the ground connection, I suggest you measure the voltage at the brake unit under load and likewise measure the voltage at the battery under the same load scenario. If there is a significant potential difference between those two points then line resistance caused a voltage drop on the line. (Small size wire or a bad connection could cause an unusual voltage drop)
You could also measure the current flow in the brake circuit to determine if sufficient current is there to operate the brake system.
In any case it appears to be a power related problem, but it is not necessarily attributed to your battery. You need to do more trouble shooting to isolate the problem source.