Mount amplified cable splitter in wall
I would not bury it inside a wall regardless of whether it violates code or not. These things will, sooner or later, need repaired/replaced. I would put in a small access panel in front of it to facilitate easier repair/replacement when the time comes. If the home is sold and this is hidden in the wall, imagine how hard it will be for the new owner to troubleshoot/find the issue.

For example: https://www.homedepot.com/p/Elmdor-10-in-x-10-in-Steel-Hinged-Metal-Wall-or-Ceiling-Access-Panel-DW10X10PC-SDL