Actually, yes, you DO qualify for federal loans as a non-degree student. You are allowed to take out loans for a period of 12 consecutive months in order to take courses required for admission into a professional program. This is how I did it - I was considered a "continuing ed" (non-degree) student at Ohio State. I had to have a financial aid counselor verify each quarter that the courses I was taking were pre-reqs for a professional program. The downside is that it can be tough to do all of your pre-reqs in 12 months.
Don't throw out the option of applying for a second bachelors degree. Just because you enroll as a degree seeking student doesn't mean that you have to actually finish it. I have a couple of friends who got federal loans for their pre-reqs via that route. The plus side is that this eliminates the 12 consecutive month requirement. The down side is that there ARE some limits on how many credit hours/quarters/semesters you can receive loans for, and if you took out loans for your original degree, you need to investigate how much/how long they will let you borrow for undergrad courses.