That would work. However, depending on the manufacturer and how the bms is used, I'd remove the bms and put a new one. Reasoning is some tool battery bms units require the tool/charger to tell them to turn on. But if your pack does not have this and you get full voltage at the outside terminals, forego that option and leave them together.
Additionally, I'd recommend putting an led on the packs connected to the bms so that if the bms fails, or it shuts down the pack, the led will light up. This may be a simple of putting the led in series with a current limiting resistor bypassing the bms (this is a guess as if this will work, but I think it will), unless it tricks the bms into not working at all.