Northern Arizona and southern Utah offer some spectacular sites for photography, too. Two can't miss places are Canyon de Chelly (pronounced "shay") and Monument Valley on the Navajo Reservation. There will be others chiming in, too.
For 3 weeks, renting will be the best deal, just because you wouldn't have the registration/insurance/etc. hassles that go with buying, especially from out of country. However, don't go overboard with the whole hog (pun intended) of a full-dressed Harley, due to your budget. However, I am not sure about a good place to suggest.
Also, where are you planning to land? (this will help determine a place to rent and route to ride)
Motels can be found for reasonable cost, but some are quite expensive depending on location and quality, but camping would be the best as far as keeping costs down. But, it sounds like you are like me, and would rather have the convenience of a motel over saving money (disclaimer-my opinion, nothing against camping before you fellow members bombard me with how outrageous I sound, etc!)