Let me see if I understand this right. You bought a prebuilt I3 system with Win 7 64bit preinstalled and the software you want to use will only work on a 32 bit system.
I have no experience with the new I systems but I've been installing 32 bit OSs on 64 bit systems all the way back to the original AMD64 chips. Win XP and Vista 32 bit will run on Intel quads such as the Q6600 and Q9300. I also ran the Win 7 beta on a Q6600 for a while with no problems. It will also run on the AMD Phenom series. Granted you're limited to 2.75 GB of memory but I have not found a problem with that. You are using a 64 bit processor but only using 32 bit of it's capacity. This is only a problem if you need to run 64 bit software. The only place I've seen it make a difference in the real world is some Boinc projects.
As a suggestion, I would install a spare HDD and try a Win 32 bit installation. Will your software run on Linux? If so then that might also be an option.