It's not so much what school your BFA is from, it's that you have it.
During your last year you will most likely have a Professional Practices course where you will be required to do at least one internship with a working Designer / Architect or Design firm. That will be the most critical part of your schooling. Push to get an internship with a prestigious firm in your area that does work which aligns with the type of work you prefer. Some firms are known for kitchen and baths, some for their overall "look", some for doing commerical or all residential design.
This can also lead to a job once you are finished.
Do not expect to earn a lot of money right off the bat. If you open your own firm be prepared to work long hours for not much money until you have a good client base.
If you want a degree with a lot of clout, apply to Parsons in New York, Cooper Hewitt, or the Chicago Art Institute. You will have to relocate to make any of those happen and they are highly competitive for admission, but the pay off will come sooner and there are varied opportunities in the design world if you come from one of those schools.
Also, here is something interesting about making it as an Interior Designer: The people that tend to do well are older women (late 40s into their 70s) and younger men (20 - 45) - just an observation over the years. Also, the wealthier the person is BEFORE they enter this field, the easier it will be for them to establish a client base.
Remember, the services you provide as an Interior Designer are a luxury item. When the economy is bad, the jobs are fewer and harder to come by.
You have to LOVE what you do! It can be a very rewarding career for the right person!
I am an Interior Designer