This is a common problem on the old BX's.
I have effected a cheap repair that will remove about 90% of the flop.
1.Remove the wipper unit, and make a note of the position of the plastic arms and washers.
2.Dismantle enough to remove the offending shaft, I took the whole thing out by grinding off the protruding pips for ease of working.
3. Use a fine hacksaw and cut a slot on one side of the bearings, this will allow the two parts of the old bushes to fall out.
4.I made up some replacement bushes from a short legth of 1/2 inch copper water pipe, (old pipe is best as it has a thicker wall).
5. Cut them to length, and then saw a slot in each one, they can then be compressed and inserted into the housing.(There is a stop in the middle).
6. Clean up the shaft, lubricate well use a new washer, and if possible a rubber O ring on the outside to excluse rain water etc.
7. Re assemble and refit..
Note!.. I drilled 4 holes in the housing and fixed it using self tapers, so it could be easily removed in the future.