If you do manage to get the cable into the stud wall there is a neat
trick for getting it out through a small hole. Get one of those
plastic covered wire thingys used to hold up net curtains and strip
off the plastic. Stretch the wire out so it forms a spiral and stuff
it into the hole. Spin it round by rolling the bit sticking out in
your palms and with any luck the cable will become twisted up in the
spiral and you can pull it out through the hole.

Saw a BT engineer do this about 20 years ago but have never had to use
it myself

HTH

Nick