Retain access to recessed electrical panel
NEC110, in general.
110.26 is the specific section covering working space and electrical space.
There are some non-negotiable minimums you have to meet.
Specifically, working space in front of the panel - 30 inches wide (need not be centered, doors must be able to be opened at least 90 degrees, covers cannot be obstructed for removal) 6 feet 6 inches high, and 3 feet forward - kept clear at all times (cannot be used for storage.)
Flush mount needs to be flush to the wall, or flush to an area of wall meeting the workspace minimum, not set back in a hole, unless the hole is 30 (or more) inches wide. Any cover must need no tools to remove (a door that does not obstruct the work space is fine - something you need a screwdriver for is NOT!)
Electrical space is the width and depth of the panel up 6 feet, or the structural ceiling, whichever comes first. That is space to be used exclusively for electrical equipment/wiring/etc.
Upon re-reading, I think you might be leaving the panel itself exposed / flush and putting removable covers over the stud bays around it, not over it - that should be fine, those can be screwed.