no, that is the charging mosfet. it is a international rectifier 3205 mosfet. after i put together what you said about the double wires to the charger and saw it connected to them i realized it was the charging mosfet, not the others on the output mounted in those heat sinks.
the fuse between the big red wire and the controller is for the battery output. we want the one that is in the charging line. it will be around 3-5A.
i called it a three terminal device because it has three legs. when you look at the mosfet from the front. there from left to right, the gate leg, the drain leg, and the source leg.
you are going to measure the voltage on the gate leg. use the 20V DC scale on the meter, put the black probe on B-, the big black wire to the battery which connects at "out-" over on the left side. that is the battery ground. i think. put the red probe on the gate leg next to where that orange diode is connected. just stick the red probe into the solder where the gate leg is soldered to the pcb.
once we know if that mosfet is active or not then we can guess more.
even though the pack is open, it is still charged so be careful not to short stuff.
for this measurement everything has to be set up normally, and you can put it on the charger to pretend to be charging when you measure.
if the charging mosfet is off we will look for one thing, and if it is on we will look for another problem. so closer to fixed, we should be able to fix it.
the mosfet doesn't look burned up. but even if it is, they are cheap and someone there will have some spares and can mail one of them to you.