I'm a Business Major myself, who just recently made the leap from IT/business consulting through to the Production Career Track. I am now working as an Associate Producer.
I believe the most important thing you need to figure out before going further is whether you like the idea
of being in the industry, or actually want to work
in the industry. I've worked in a number of different industries now, ranging from Aviation, IT, construction, and its obvious to me that the games industry is the most cut-throat and tough industry I've seen. You say your university path won't allow you to go the other tracks, but what do YOU want to do? If I have learned anything it is that you need to follow your HEART not your OPTIONs. When I started looking for a job in the industry I had no doubt regarding what I wanted to do, and I think that really helped. Choose the area that you most want to be in, and you'll put the most into it.
When I decided I wanted to get involved, and not as a programmer or artist, I spent about 6 months just researching and considering whether it was really what I wanted. Then I found the job type that I really wanted (Producer) and researched that to make sure I would be happy with actually being a Producer. On reflection, being inside the games industry is a very different experience than what I thought it would be. Games are a serious business, costing serious money, and require people seriously committed to them as a career.
Most of the "business" people I have come across in the industry so far are ones from within. Programmers/Artists who started their own studios in partnership with others and such. Our HR manager is the exception, although obviously there is little about his job that has any direct involvement with game development.
The pay is terrible, the hours are long, but the work is the most fulfilling I have ever done. I don't mind staying to work longer hours, as the team dynamics are fantastic and theres a real buzz around each project. Its an industry of passion, and its the passionate people that get hired. There is no real opportunity for studio's to make mistakes or take on risk, as one failed AAA title can put a studio out of business. Hence, entry into the industry is problematic.
I'd recommend checking out the Producer track, be aware that its not "typically" an entry level area, and the AP jobs are fiercly sought after. Good luck and hope the study goes well