Okay, an update. Download worked fine, game installed, but would not start up without saying "File is corrupted" when it tried to do the Zero Day Product check via Games for Windows Live. I looked this up and it was very common with Bulletstorm, so lots of people had already worked out what was going on. Apparently, there is a program file zdp.exe in your install directory that requires more permissions than the standard user. I ran that as administrator and the game executable as administrator and the game worked after that. Updating via GfWL and starting up, game plays afterward fine. Seems smoother, too, though it didn't let me get very far without updating.
I contacted EA Origin customer service to try and get the game attached to my Origin account as per this reference:
http://forum.ea.com/eaforum/posts/list/5309323.page#12650912
Just in case anyone else wants to give it a shot. I had no luck with the CS reps as they all were clueless. One of them told me Bulletstorm had no digital version. Another told me I needed to provide manual and key scanned image. I advised them of the post, but neither seemed willing to look into it beyond to see the key is in a different format than their typical key, which means they can't validate it in the typical way.
So I PM'ed one of the Origin developers--the guys suggesting contacting CS--to get more info. Kinda annoying that there isn't some consistency, but whatever. The Gamer's Gate way allows for an easy backup of the files.
I write this in case anyone else runs across these problems, maybe this will help. Origin's still got a long way to go before it has the awesome quality of the Steam "experience." Really kinda sad they have so much trouble getting a 2011 game to work with the platform.