I have a 1990 Burley Bongo. Your rear BB when new was put in with a friction fit and held there by snap ring clips. There was also some what looked like epoxy. I suggest you clean it up as best as possible and see what is going on with it. I have never tried to remove mine but I have removed the outer bearing seals with a jewelers screw driver and worked grease into the bearings. I am guessing the original owner may have tried to work on that BB and gave up. If the bearings are staying in and the axle is moving to one side then you should be able to put a new one in. Hear is a listing for a new one.
Burley Bottom Brackets Good luck