Traveled by air several times with hunting rifles and sidearms and I agree 100% to list these only as air guns, not guns, there are specific rules for both.
Asking is essential to quick check in/out, touch that case when not given permission to do can lead to criminal prosecution.
Allow me to illustrate: I fly from the south to Wyoming with a "gun" but a sudden storm comes up and I'm diverted to NYC.
If you touch that case in NYC you could be guilty of several misyermeinors and perhaps felonys.
Bottom line, call the TSA at (866) 289-9673 and get a case number and ask your questions of them before hand. Nobody here can give you legal advice, you need to seek that out yourself.