If you have had 6 genuinely good years together and if this is honestly out of character then I can understand that you don't want to get rid of him. Only you know whether that's really the case.
But he can't be around your baby while he's like this and he shouldn't be around his older child either.
For now, you need to continue focusing on you and your baby. You have your arms full, you can't carry him too.
He needs to stop drinking. Until he can do that, he can't have a safe relationship with your child, or with you. He can't sort his MH issues out while he's drinking. He probably also needs to look for another job that is not night shifts, they are notoriously bad for both mental and physical health.
it was triggered because they fell out and she moved out with his son. There was no MH involved on that situation.
Why did they fall out? Do you know there were no MH issues or is this just what he told you? How old was his son when they split?
From what you have written it's possible this is an isolated period of mental ill health and you may be able to have many more good years together, if he does the work of sorting himself out.
It's also possible that this is just what he's like and you've been ignoring the signs for a very long time.
Even if this is an isolated illness, you may never be able to forgive him. You may never be able to trust him again.
Time will tell.
For the next few months, just concentrate on enjoying your lovely baby in a safe environment. Don't ever let him see her if he's been drinking and don't let him see her unsupervised until you are certain that he has properly stopped drinking and his mental health is stable.
Also let his son's mother know what's going on, regardless of what kind of relationship you have with her. She needs to be able to safeguard her child as well.