Personally, I would listen to the wisdom of your mother. She lived this, she walked in your current shoes years ago, she will not steer you wrong.
And IMHO, he is not REALLY trying to stay sober. He is simply pacifying you.
I would keep the focus on myself and beautiful baby.
There is nothing you can say or do to help your BF. This is his disease, he gets to own it, and he gets to choose if he wants embrace recovery.
I certainly would encourage you to educate yourself about addiction. Read the stickies at the top of the forum, alot of great information and support.
We are here for you, you are not alone.
Neither you or your child deserve to live with an addict. You both are worthy of so much more.