I don't have a BSN either. I went to a NP program that didnt' require a BSN.
However, I did have to apply twice. The first time, the interviwer peered over his glasses at me and said "So, have you taken any HARD classes lately?" (My MSN was only 3 years old at the time)
When I got rejected, the admissions committee recommended that I take upper lever undergrad or graduate level basic science courses, and try again. I did, got in and now am in the top of my class. (most rejection letters don't contain recommendations, this school is unique in that regard)
Lesson: There is a general tendency in the MD community to assume that nurses aren't as smart as MD (after all, if you were smart enough, you would have gone to med school in the first place, wouldn't you?)
Not that a BSN will keep you out. I dont' think you need another BS. But I strongly recommend taking basic science classes (examples: biochemistry, microbiology [not the kind you take in nursing school, the kind that a biology major would take], virology, physiology [again, the biology major kind], genetics, cell biology, molecular biology, etc. Not only will this help "prove" you are "smart enough", it will also be a big help to have some exposure to these things prior to starting medical school. I took a total of 6 of these...2 in the summer, 2 in the fall, and 2 in the spring (I interiewed for the second time in the middle of the spring semester, and got in.) Plus I worked 2 part time jobs. It was hard, but it paid off.
One thing that I didn't do that I wish I had done. Call the admissions department at your closest med school and make an appointment to talk to someone about your credentials and what would improve them. You need to do this BEFORE you apply, othewise, they wont' see you. Go in full interview regalia, bring your transcripts and MCAT scores. Many, many schools have someone who will meet with you, discuss your chances and make recommendations. If I had done this before I applied the first time, I could have saved myself some time and money.
Good luck