I think the best way in would be to remove the " cloth " centre trim, this is only glued on.
Then cut a hole behind where this fits big enough to get your hand in. You should then be able to reach up and grab the Bowden cable. Might have to force this cable out of the internal handle latch but worse comes to worse you can cut this cable so that you expose the centre wire to give it a tug.
Afterwards you can just refix the centre panel with glue or tape to cover the hole.
Only snag I can think of is the metal work of the door behind the stick on bit. I can't recall how much of it there is, you may have have to cut a bigger hole than is actually needed.
Good news is that the Bowden cable is on the card side of the door and so even if there is solid metal behind the hole you may be able to squeeze your arm in to get at things Worse case scenario is that you'll need to get a new card from somewhere.
You could be even luckier. I suspect that what may have happened is that the lever that the button screws on to is out of place, it's floppy without the button guiding it through the hole, so that this is just jamming against the top of the card when you try to unlock. You may, if you are a very lucky man, be able to " fish " this into place through the button hole. If not you should be able to reach it from the hole you have just cut !
Either way, it sounds like a lotta lotta fun ! Good luck.
Another idea for you. Depending on the way it was fitted you may be able to remove the big speaker from the card. This will give you a big hole to work through and with a bit of luck and contortion you MAY be able to reach the internal handle mechanism.