Shopping for something that can be livable in the winter

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.

And also get some Damp-Rid, they tend to work very well!

TTs with less than 4 season capability can be made for 4 with some thought and work. You have to start with a preferably wood framed unit with enclosed underbelly, this was what we made into a -45 rig. From a 3 season . Wintered over in Ontario, Manitoba, Alberta. Never froze .

Dropped the belly, heattraced all water lines placed waterbed heaters on under the tanks. Power for them was run up to where we could either plug into 120 or inverter so that they could be powered parked or on the road as long as the truck was running. Screwed pvc 'brickmoulding ' to the frame so we could add 1 1/2 foam insulation then replaced the belly. if the axles hit the frame extension , so what. its pvc and coroplast. A little dent

Single pane windows were dealt with by cutting 1/4" plexiglass to fit, remove handles- trimming them with foam weatherstrip, held them in with turn mirror clips.

Refridgerator lower vents must be restricted when temps drop below freezing, foam pipe insulation works well, it is easily stuffed into the louvers, taken out when temps warm up. To keep the fridge working, the outdoor coils must be kept above freezing, below 100f or so. A daily chore dependent on weather,

It is critical to ventilate moisture, use a dehumidifier to reduce condensation in the walls and on windows.
Even the best touted mfgr 4 season units cant compare with this .

Take a look at this YouTube link to get an idea of what COLD oil field workers are up against in N. Dakota:

RV's In Extremely Cold Temperatures

Watch these videos for tips and hints as to cold weather in an RV.

Also take a look at Nash branded 4 season trailer, though I doubt you will find one on that budget. I considered bidding on one that was used as a FEMA trailer after Katrina a few years back it went for $6,800 at auction for a 25 ft model in good condition, when un-issued standard FEMA trailers were selling for $2,000-$2,500.

$7,500 for a quality four season RV good to minus 30 isn’t going to happen.

Arctic Fox is a good choice for complete 4 season camping.
Beware of any RV that claims 4 season and then has exposed wet sewer connections. Valves should be installed to keep any liquid inside the underbelly.

this was a fun read that might shed some light on your question:

http://www.trailerlife.com/lifestyle/braving-the-yukon/

Gosh NO!
That is basically a 2 season RV. What you need to look for is something like a Teton. They are really really heavy. Being constructed of 2X4s with appropriate insulation. But it is highly unlikely you would even find one of those for your price.