The clinometer makes a sort of sense - as does an Engineer's case for one of the many measuring devices associated with them.
What doesn't make sense is the use of a hat badge as a decoration - and that's what I believe it is - a simple leather case, that someone put an attractive patriotic-looking badge on - perhaps for a play - perhaps because it looked pretty.
Just so's you know - if it's military - the leather will generally be of a 'robust' construction for continued service, and there will be stampings of initials on the case body, as well as on the leather straps - indicating who made it, and who inspected it.
In this case, I doubt they'll be found.
Also - Russet leather's associated with post-1900 equipments, while black indicates earlier service, and we stopped placing metal devices on things after the Civil War - finding that embossing worked nicely, and didn't add to a greater cost like accouterment plates did.
It most closely resembles the case for a 'D-TEX' Watchman's Clock - the sort of thing that a Night Watchman carried when making the rounds and inserting a key that embossed a tape that reflected the time it was embossed - ensuring that the watchman was making his rounds in a diligent manner.
Scouts Out!