Barrel plugs are a poor choice for this application. XT30 or XT60 would be better.
If you can open the pack, there should be someplace you can disconnect things. On cheap ones, this will be a soldered connection. A typical BMS board will have spots labeled B+, P+, B-, P-, C- and a balance wire plug with one wire for each cell. B connections go to the actual cells. P connections (Pack), go to the load. C- is charger. All the + connections are just joined together. I recommend disconnecting the B+ connection first, then unplug the balance wires from the BMS. The B- connection can stay connected.
Once you have the BMS unplugged, you can use an ohmmeter to measure from the negative charger input to the B- connection. Reverse the meter probes and measure both ways. This should look like a diode. If it measures open circuit or short circuit both ways, then the BMS is likely toast.
After repairing the charger plug, reinsert the BMS balance plug and try to make the negative end of the connector go in first (angle slightly). Make the B+ connection last. After reconnecting the BMS, it's a good idea to measure voltage across each of the balance resistors to make sure none of them are stuck on. All should be zero volts.
If you're not sure, post pictures of your pack insides.
"One test is worth a thousand opinions"