Your budget is entirely reasonable, even generous. You could stay in a decent (but not fancy) motel every night for that with a bit of care.
If you have an automatic transmission, a transmission cooler would be a great idea before towing. For a manual, it doesn't make as much of a difference as the transmission by nature doesn't generate as much heat under load.
A teardrop-style trailer is a great idea, and is easier to set up in bad weather. A small pop-up would also be reasonable. A normal tent and appurtenances is a third option, and probably the least expensive (but a little more hassle). In any case, it sounds like a good time to me.
