HI, Welcome!
In regards to the question, it would be easy to hack the buttons, just disconnect them at any point in the node, then solder your conn in,
however , if there are other buttons running off that node you will also need to not only cut them out as well, but you will need to add a bypass wire connecting them back up , outside of your button circuit
in the back of board image , we cannot see the nodes (the pcb connections), so cannot specify exactly what to do.....
...also adding a flip-flop chip will remember which button you pushed last .. if bluetooth can't do that on its own....
I fear thou that is the easy part, i wonder how you are going to tie bluetooth in to radio, and what is the bluetooth going to be doing? is this prebuilt, ...ect