TL;DR
You do you, but consider that a trip like that is gonna be taxing on both your scooter and yourself.
Hey friend! I ain't in the habit of telling folks not to do shit they wanna do, so if you're set on Scooterquest then make it happen.
That said, if I was planning in taking a 50cc moped from Mesa to socal, these are two things I'd want to put some serious thought into:
1. I'm not a mechanic, but little commuter mopeds like that couldnt of been designed with sustained travel in mind. If I had to try to take one on a 350 mile (and then some, because if you're back-roading it you're not taking the most direct route) haul, I'd definitely be concerned about breaking down.
2. You're planning a multi-day endurance ride through southern arizona rolling into the summer months, and your scooter probably doesnt go much faster than 30 (if you're riding the red line the whole time, which can't be good for the engine, see point 1). You're gonna be out in the arizona summer for days, even assuming everything goes smoothly, and I'd wanna be damn sure I'm equipped to handle that before attempting it.