Look where the joists meet the ridge beam where you're going to put the door.
Get some dimensional lumber about 4 feet long, at least as thick as the existing beam. Nailing a couple of 2x6's together would probably suffice in the short term.
Give yourself enough room to work, and using a jack post and this 4 foot beam segment, support the joists in the area where you are going to work. Don't over support, as you'll lift the floor above. The idea is that you want to take the weight off the wall where you're working. Jack the post until it's snug, and take out one wall stud in the middle of the door. Check the supported joists for level, and jack up if necessary.
If required, remove another stud, and adjust the jack again.
Put in your door frame and header, and slowly jack the temporary beam down.
It has wheels. Do you have any plans to move this thing? EVER? Because trailers flex, and there are two strategies for dealing with that.
#1 make careful material choices so the trailer structure can flex without taking damage. That's where drywall and mud may not be your best choice, for instance.
#2 make the trailer base and structure, so rigid and strong that it forces all the flex into the suspension. You might think #2 happens automatically, but it does not. Either way, you have to deal with the forces at hand.
The reason for thin walls is simple: it adds square footage. If motion is an issue, I'd say "think like an airplane not a house". Airplanes are insulated.
Unfortunately, the simple-living and alternative-energy communities are often hijacked by the "I wanna live in squalor", "gov't off my lawn" and "off-grid, like the unabomber" fringes. Look at any of the "government shutting down timy-homes/off-grid tech" and it's actually the latter they're concerned with: drinking downspout-water or not having smoke detectors in a house full of kerosene heaters, which is usually about being "cheap, the bad way".
And in your town, you will be one of the people swaying the government on the question of whether tiny-houses are good or bad. My advice is be a good ambassador: don't be cheap, be classy. Be the magazine showpiece tiny home, which is done less in dollars than in time, care and pride. (well old-pride, when it was a motivator to excellence, not new-pride, which is puffery instead of work.)
Best Answer
I would suggest that you brace as required to keep the structure straight and square. The sidewalls will be affected by how straight the lumber is in the first place and also by the process of installing the side wall sheathing. You will need bracing as required to push the shape of the framing into plumb and square and then hold it there.
Also be aware that since the integrity of the structure is not complete until the roof trusses are placed and fastened to the upper plate of the sidewalls you will need bracing in place to deal with forces of nature. As you begin to install the side wall sheathing even moderate amounts of wind can place significant pressure on the wall structures so bracing will be needed to make sure that a strong gust does not come by and flatten all your work.