The Permatex Ultra copper will work very well, but won't be completely permanent. You'll probably need to re-do in a year or so, depending on your actual usage, maximum heat and how much it is exposed to weather, direct sunlight, etc. but it will be worth it.
Really clean the contact surface where you want the sealer to stick with a non-oily cleaner (alcohol, acetone, etc. but NOT mineral spirits, paint thinner, etc.) let it dry and apply the sealer. Cover the sealer with some waxed paper (smoothly applied, covering the door opening, too) and then install or close the door(s). Let it dry overnight. Remove or open the door(s) and gently peal off the waxed paper. If some of the paper sticks, don't sweat it, it'll dry out and/or burn off with use. That's why you use waxed paper instead of plastic wrap. The bits of plastic wrap that stick will burn, stink, smoke, etc. (don't ask!)
Good luck and don't forget the pictures...
My first try was with the High Heat BLACK sealer 'cuz it looks better on a smoker, but it only lasts a few cooks...good choice on the Ultra Copper.