How to remove solder bridges on a 6-layer PCB - how to rework a board that won't heat?
I usually use a hot air rework tool at the bottom of the board, you would be surprised that even getting the temperature of the board to 80C to 120C can do wonders for soldering by heating up areas of the board that are difficult. (If you increase to 120C on the other side it will cut the heat leak in half)
Sometimes this doesn't always work, so we got a infrared rework station and use that to heat the other side of the board.
Make sure you have good flux (I use this, it seems to work much better to wet solders and break down the oxide layer)
Another thing is if using an iron, use a new tip or a very very clean one as these build up oxides and reduce heat transfer.
(You can drill holes in a 4 or 6 layer board, you have to be really careful and not short layers together, a microscope is needed to inspect the hole if you need to disconnect a mid layer trace, this only works if there are planes or nothing between the fix and the outer layer of the board)