For one person living, I'd suggest to envision the smallest size livable. A smaller rig will need less heat and is easier to reach comfortability.
A few years back, I was looking at the possibility of Wintering in the Bakken oil fields in ND, living in a trailer. I quickly discovered that my thoughts about the un-livability of nearly all TT's was well founded. At that point, I knew that extensive modification was necessary. I also knew that I wouldn't have the heart to tear apart even a fairly good rig.
I searched for something used in a convenient size that was cheap and had a good frame and running gear. I ended up with the trailer in my signature line. The thread that is linked explains my choices but Photobucket has ransomed the images.
Short summary: Tear down from inside, rewire, replumb, and insulate to the max using the dimensions available. Replace high electrical draw furnace with propane space heater. Replace windows with Low-E sliders. Use thermal breaks everywhere possible, the most being the wall paneling as that is where the most conductive heat loss appears as the studs can transfer quite a bit of cold to the interior.
I'm pretty happy with the results and the last task, insulating the waste tank and connected drain plumbing, is the only bit left undone. I have the materials here but my venture in ND never materialized. I'll get to it to finish my polar build some day.
If you plan to park for periods of a few months, having a tight skirting system is going to be the most benefit. Heating that space underneath with lamps or electric heaters will be the #1 thing you can do for comfort. Plumbing can be made to work although it will be an ongoing chore. Any gap in insulation or failure of heat tapes will need days of recovery to unthaw at the lower extreme temps. Waste will need to be dealt with and that situation is unknown.
Good luck with whatever you find. FWIW, my total costs for the modifications was about $4500. I did have a few things on hand. I also added a solar charging system and cost for that with 120V inverter was around $1500. Sale price of the Starcraft was $750. These totals line up with your budget but know that I spent nearly 6 months of daily work on the trailer.
