I wouldn't attempt it. Something like contact cement or white glue may be easier to remove, and other types of glue have been discussed in earlier posts, but the risk of damage to the "motor" exists. Cheaper MMs don't have delicate coils attached to the cantilever, and many generic MM stylus came right out of the compliance and stylus to work with.
I would consider (unless you know the cartridge rebuilder) the home rebuilding craft for situations where it keeps costing too much money, or if you have the cartridge to "hack" with no costs. Shure and A/Ts are found at garage sales for almost nothing.
I assume a rebuild would consist of the coils too, and it shouldn't matter what you do, but "send away" cartridge repair may require some original measurements, etc. Cleaning up another's mess is not always easy. It can be costly and risky.
The reason I would repair the original myself is because there is a good chance the original factory work would never be outdone by a second rebuild.

