Well I solved it. After a few tries with EA's origin website, I managed to add the product key for RA3 and it started downloading the game again. I copied the primary data folder from my steam folder to the origin folder (I only wanted to transfer the core files to minimize risk of transferring over something that caused problems), and after resuming download it jumped to 97% and completed shortly thereafter.
The only major difference I can see is that it didn't have to install this special file, I think it was some distributable version of C++ or something. Anyway either as a result of that or something else, running the game through Origin worked fine. As a side benefit my computer didn't crash when running other games, so I think that installation of C++ made my computer pretty unstable. Good thing restore points exist!
I hope my experience helps someone else. So far I'm close to finishing the first soviet campaign and the game runs nice and smoothly.
-Rad.
