Are your blood tests IgG tests? That's important to find out. If it's an IgM test, that's the wrong test.
If they are iGg tests, then I think you should get a Western Blot test. That's a different kind of test. The hsv2 IgG misses 8% of infections, and the Western Blot will confirm the IgG, or the culture.
If you have it, it's not surprising your bf hasn't gotten it.
Ghsv2 transmission, female to male, over the course of a year, assuming sex 2-3 times a week:
Only avoiding sex during an outbreak - 4-5%
Adding condoms OR daily suppression - 2-3%
Adding condoms AND daily suppression - 1-2%
So given that if you're just avoiding sex during an outbreak (and you haven't had any), there's a 95-96% chance he wouldn't get it from you over the course of a year, if you do have it. (You don't mention if you use condoms, so for now, I'm assuming you don't.)
Your doctor is giving you strange info. If you don't have herpes, why would you have a positive culture? Is he assuming you had a false positive culture? That can happen, though it's rare. But people don't just get one herpes outbreak, and then don't have herpes.
Your doctor can order the Western Blot test. If he won't, and many aren't familiar with it, you can go through Terri Warren on her site - https://westoverheights.com/getting-a-herpes-western-blot/
So first, get copies of your lab work and blood tests, and we'll go from there. Ask your doctor for it - you have the rights to have copies of those.