I think you have quite a lot of options, actually. Many dojo will allow a visitor to stay at the dojo either by sleeping on the mat (bring your own sleeping bag) or by supplying a guest room (like our dojo).
Also, many cities have hostels and such that are really cheap at around $10/night. You can go to an aikido-rich area like the San Francisco Bay Area and visit a different dojo every single night if you want, all within an hour's drive.
You can also try contacting someone at a prospective dojo to see if there is anyone there who is willing to host you for a couple of weeks. I've done that before, too.
Good luck! Let us know how your trip organization and such goes.