Further to this, I tried again last night with pencil tip soldering iron bit. It melted the solder, but I couldn't get it out. I think I'll buy a solder sucker and try again with that at some point in the future with the board upright, heating one side and sucking the other side.
Since I don't need those GPIO ports at the moment, it seems foolish to risk the Pi's health any further at this point.
P2 and P3 are completely filled on this Pi as well, but I'm not sure what they're even for.
At least P6 is still free of solder so I can put a 2 pin header on for a reset circuit if I want.
The other thing I noticed when trying to "trial fit" a 6 pin header is that if you put one on the topside, it will make it very hard to use a ribbon cable on the main GPIO header. I know it's designed so that P5 would go on the underside, but that's no good for my case. Maybe it's time to get creative and just bung some wires in there? (But that will look awful LOL) Or maybe a header with bent or offset pins?