I suggest you abandon the idea of mounting between the studs. Most large panels (large=high port count) are designed for a 19" rack (or a 23" rack - Telco). Most "panels" that would fit in a ~14.75" space are 4, 8, or 12 port verticals.
What would, IMO, suit you better is a wall-mount shallow rack. You can get them ~four or eight rack units or longer (up to ~12 Rack Units). Then you can put in a pair of 24 port UTP panels (or a single 48) and a "multimedia" panel to handle the coax and whatever other media type you may install in the future.
Most wall-mount racks have either a drop-down or swing-out feature to make punching and troubleshooting the cable much easier.
"trust me" ... this should work much better.
Good Luck, Happy Holidays
Scott