Your August/September (September so kids are back in school, but maybe you need to be there during break) timing sounds ideal for a camper-van, depending on how much you want to see on the way. I don't know much about it--- current rental market, pricing, etc, but know there are sites where individual owners will rent out their personal motorhomes/vans for trips. We did rent a converted Dodge Caravan for 5 days from a company called Jucy, a few years ago, and it was slightly more expensive than a car, but much cheaper than car + hotel.
Also, how much of your trip is in either of your two destinations vs the travel between them? If you're just looking for a way to arrive at the destination, have you looked at pricing to rent a car for just that leg of the trip and then turn it in when you get there? Stay a week or whatever amount of time, then rent another car for the next leg of the drive?
If you can buy a reliable car for 2k, I'd like to think you can resell for same price (after vacuuming out the dog hair) when you return. Have it checked by a mechanic before going through with the purchase.