How do I remove a plunger inside the overflow assembly? without the rod?
Soak it overnight with CLR to soften any lime/scale (the bucket, i.e. stopper will often have crusty stuff from years of just sitting there) then douche it with penetrating oil like liquid wrench. Then use a stiff heavy-gauge wire bent into a hook shape to try to fish the bucket out.
Then, instead of trying to replace the bucket and wire assembly, leave it out and put a blank trim plate on the overflow. Hand them a rubber stopper and scram before there are additional problems.