Step one of rebuilding the e-bike my co-worker gave me is grafting a different battery management board onto the battery pack.
Transplant complete!

I still need to figure out how to wire up the charger plug. They used an XLR connector with four pins, and the charger doesn't turn the power on until it senses something (maybe a temperature sensor?)

Edit: i just needed to check the back of the charger for the pinout

A lot of fiddling later and i have the charging port connected and everything stuffed back in the original case.
Good news! It charges!
It's Friday night hacker meetup and i am working on reviving the e-bike with the parts i ordered from AliExpress
It's gonna work!
That took hours longer than it should have because of dumb and fiddly little problems, but it's a working e-bike now. I just took it out and got it to to 24mph. That's faster than i like!