I bought a 1997 K2500 4x4 Suburban ten years ago on E-Bay for $8300.
So....it was 10 years old then, and had 100,000 miles.
Flew to Albuquerque, NM., picked it up, and drove it home.
Today it is 20 years old, and has 130,000 miles
It is an absolute brute!
I LOVE IT!
Big brakes, and 454 V-8.
For towing, I use it maybe 2 or 3 times each year. Towing use is similar to your situation, once 250 miles each way, and usually a short trip or two of 20 miles or less.
I tow a (14,000 lb. GVWR) 20' equipment trailer (electric brakes), with my medium size Ford TLB aboard. Total trailer & tractor weight is exactly 12,000 lbs.
Some here may say that is too much weight, and perhaps it is.
I did add "Airlift" air bags, and I also use an "Equalizer" brand 14,000 lb. weight distribution hitch.
Bottom line: It tows GREAT, and I keep substantial distance between me, and the vehicle ahead.
I do think your budget should be increased to $5000, to get a reasonably reliable vehicle for your described use.