I'm not sure if I understand the question correctly... Mainly, drill the mushroomed end of the rivet with a drill bit that is slightly larger than the shaft size of the rivet and remove the cover. Then, you should have easy access to what you need to do.
We use a heated 'solder sucker' and that seems to work very well with thru-hole parts. We don't do that much volume on repairs (high mix, low volume), but it does save a tremendous amount of time when trying to do thru-hole by having the solder sucker just take everything out of the hole, remove part, insert part and resolder.
If this is completely off the mark, please, provide a few more details.