Hi Rachael,
It doesn't sound like a bad set up to me. I grew annuals and tomatoes last year under a four ft grow light upstairs in my attic, with only on small westerly window to aide the lights.
I really didn't get as much light as you probably will. I still had tomatoes though. You will probably have to hand pollinate them. But some of mine I didn't and I still got fruit. I'm not sure how that happenend, it surely got there somehow.. anyway..The biggest problem I had was spider mites, and my tomatoes trying to come out of the pots. (I chose to small of a pot). as well as gnat larvae and mildew. I fought it all winter long. And it was LOOOONNGGG! The main problem for the tomatoes were the spider mites and the gnat larve. But the regular use of fan on low and frequent day time misting, and a dose of course sand on the top of the soil helped too. Plus I had to fight the urge to over water them constantly. But I definetly think your situation is very do-able. It sounds like it is much better than mine was. Good Luck!!
Melody