Desolder the header pins. Desolder while the screws and mounting hardware in the next two steps are still attached, so the board is more stable.

Look straight down and check that there most of the solder between the pin and the sidewall is removed. Push each pin sideways to make sure it is detached.

If it looks mostly clear but the pin is still attached, try pushing the pin to the opposite wall (where it is clearly detached) with a moderate amount of force. (Do not force it too much though or you risk damaging the through hole plating)

If it doesn't budge with a moderate amount of force, try using the iron to push on the pin the same way. If it attaches to another part, then the solder isn't sucked up enough.

Optionally, you may want to mod the PCB to use a socket to connect between the top and bottom, so it will be easier to access in the future. You will need to replace or cut short the header pins. It uses 2.0mm pitch 2x7 headers. There is approximately 7.2mm between the boards. I used Molex 87758-1450 pins and Amphenol 63453-114LF socket.