I wouldn't do it. My Dad's friend took his Beretta case to Canada a couple years ago and it got broken and it was a huge hassle. My brother also had an issue the first time he tried to use his Beretta case. They are not very tough and crack along the hinge.
Get yourself a Pelican case and put some real padlocks on it -- not TSA approved (hey are all too small and flimsy).
After you open the case at the airline counter to prove it is not loaded and get the little card put inside, lock it up and wrap the locks with duct tape to keep them from flopping and torquing the lock holes on the case.
Send it off to TSA and don't worry about it. I have traveled this way with my Pelican case to Canada every year for the past 17 years. I have also flown up to Rhode Island for sea ducks, SD and Kansas for pheasants, San Diego and Phoenix for quail. My Dad and brother now do the same thing.