Hi. Welcome to the forums. I'm new here myself but here is my take...
I'd be very, very concerned that your BF doesn't see this as a problem. I've learned (from my AH) that addictions are progressive. Each time one quits and relapses it's worse than it was the last time. If your BF isn't willing to admit he has a problem AND get help it's quite possible that your best days in this relationship are behind you. I can't tell you what's right for you, but, were it not for the most amazing kid my husband helped create, I wish I would have walked away from him as soon as I had any inclination he was abusing alcohol. It would have saved me so much pain, money and anger.
Wishing you peace and clarity.