Personally, it would drive me nuts to have to lock and unlock the door with a key each time, because unless you install self closing hinges the door will not stay closed on its own. going in and out with stuff in your arms would be a pain. Not to mention the likely hood of someone breaking the key off in the lock. Standard house door frames are probably not the best for mobile use as they can be hard to seal around, and mounting and trimming can be a pain depending on the construction type of your box.
There are doors made for mobile home use that have an aluminum frame that screws in through the exterior face similar to a rv door with putty installed between door frame and exterior wall. We use them on many of the mobile construction office trailers we build. Might be a cheaper alternative to an rv door. Available in tons of sizes and with and without screened windows. Kinro is one of the bigger manufactures. They come prehung and drilled for knobs. if you want to stay with the deadbolt you could either install two or install a blank cover over one of the holes. Here is a link to a site that sells them.
34" x 80" Kinro Blank Out-Swing Exterior Door: Mobile Home Parts Store
Another option is one of the many RV surplus stores in Elkhart. They had lots of doors to choose from when I was there in May.