I gutted and rebuilt my trailer, fully insulated.
Here is what is inside the aluminum siding: 1" spun fiberglass (compressed) against the aluminum siding and around any wiring, 1" extruded polystyrene friction fit between frame members, 4 mil poly vapor barrier, 1/4" fanfold extruded polystyrene, lastly-paneling.
I chose foam board instead of spraying because of cost. I chose the spun fiberglass against the siding for moisture abatement and ease around wires. The fanfold insulation behind the paneling allows a thermal break between the frame and the interior paneling.
If you do spray, hire a foam installer. It will be cheaper and easier than DIY. Allow for conduit or some type of wire runs to be accessed after spraying (if you forget one wire or device location, you'll be chiseling foam to get the wire into the wall). Install junction boxes that are covered by an access plate for any future electrical upgrades/diagnosis of problem (I used a light fixture-combo-access panel).
Tyvek on the outside of the frame will allow for less convective heat transfer but may hinder moisture migration out of the frame. Your call but I reasoned that a potentially leaky trailer might need some way for water to get out of the wall. so didn't seal the frame space on the outside.
I also replaced almost every window with a triple pane, Low-E glass unit. Results--In Summer, if I let in cooler night air through windows (75f?) and close windows, the trailer temp will be 80f around noon with an outside ambient of 90+f. In Winter, at outside ambient 25f, inside trailer 35f, I can raise the temperature to 78f in around 20 minutes with a 1200W electric heater. It will take a few hours to heat all the mass inside the trailer but that is with a fully cold soaked trailer.
Good luck on your project. If I had one suggestion it would be to draw plans for mechanicals and fixture placement. They don't have to be architectural drawings, just to act as guides and reminders of what the schedule should be for construction. Also, don't be afraid about plumbing. I plumbed the whole trailer using PEX pipe/ABS with the best fittings I could get for less than $200.
