How to determine voltage/amps a circuit board expects
I have never used that board or a JST splitter, but I am going to make a guess:


If the batteries are in two different places, that probably means the board has two different circuits that do not connect to each other.
If it overheats when both batteries are together, you are probably applying twice the needed voltage to one of the circuits and none or too little to the other.
Your splitter doesn't say it splits the current evenly. I assume it isn't.
In order to use one battery pack, you need a circuit board that makes sure equal power goes to each side. What I would do is buy two single battery mounts and place them next to each other, then run separate wires to each side.