.....well.........now you've asked something..........never considered this before LOL
Expanding foam would be possible, but, you would have to guage the EXACT amount........or it could possibly blow your plasterboard off................ :o(((
Also, it would take loads of it ...........you'd probably have to hire the kit if it's available or it would take a lot of cans.
What about the teams who inject cavity wall insulation............they check it out for you as a free service............small holes in the board, just big enough to get a nozzle in.........see if it's possible.
Best of luck