The only one I know of that easily breaks down wheels and all is one of the Bennington models, and maybe a bellcrown? Both are over your budget.
What you will find in your price range is a nice used road cat, that will not break down whatsoever… Most of your metal carts, pacifics, bellcrowns, benningtons, sprints, all have adjustable shafts that remove… However, how you would secure one without scratching it to pieces or move it easily without shafts to help, well good luck with that…
There are cart racks you can install on the rear of a trailer, or if you’ve got a bumper pull you can load it in the truck bed.
Sprints are very new to the market and close to 3k, not sure you’ll find a used one for a while. Pacifics and bellcrowns are going to be 4k ish with benningtons and kutzman makes a nice cart if you order it will be 4k+.
You just aren’t going to find adjustable/removeable in your price range, but lots of pretty road carts out there that will do the job. You just have to get inventive with packing them in.
Wheel width will vary depending on construction of cart and the shape of shafts. Ie, are the curved shafts, does the box sit higher or lower to the axle, so you can’t really size a cart for a horse by the wheels. I’m not sure on shaft length for you by blanket size. You may fit into a “horse” or a “warmblood” size vehicle. Any of your metal carts mentioned above have multiple adjustments to suit multiple horses, a wood cart does not.
My advice, save up a little more money and get what you really want/need cause it will only frustrate you otherwise and you’ll lose money buying and selling.
Any of the carts are suitable for training level CDEs, beyond that you’ll want 4 wheels for marathon. Though I would keep the cart for cones and dressage, lighter vehicle will make dressage a little easier for the horse to show themselves off better. Many do all 3 phases in their marathon carriage though cause they can’t fit two vehicles to travel.