I took a quick look at the Penn. site, and it looks like you can't do it without a Penn. license or ID card. I didn't look any further, but you might: for example, with your visa, address and job can you get a Penn. license? With a local license you can probably insure and register a bike. Note, however, my use of the word "probably."
There are certainly states where this can be easily done, but the ones I know of are not near you--Florida, Montana, Washington, California. If you get stuck, give that some thought.
On the other hand, it's not necessary to actually own the bike in order to ride cross country. Someone else could own it--just make sure you're covered by whatever insurance applies, and come to whatever arrangement you like with the (actual or paper) owner. This part would work differently if you wanted to sell the bike or take it into Mexico or Central America.