Why don't you think you deserve to be happy? I think that feeling unhappy is a legitimate reason to leave a relationship. In your situation, you aren't just feeling a little bit unhappy, you are desperately unhappy, stressed and ill.
You seem to be positioning leaving due to your unhappiness as the selfish thing to do. As you have seen, every single person who has replied on your thread (I think) think that is wrong.
You can't see the damage being around him is doing to them because you grew up with similar/worse and think its normal to see an adult behave as your partner does. The most obvious damage it is doing is making his behaviour normal/acceptable to your children so that they replicate either your relationship or his (appalling) coping methods in thier own lives. You want better for them than the life you and P are living at the moment, don't you?
I was going to write out a list of ways that you and your children's lives would be better if you weren't living with someone who is unable or unwilling to act like an adult with cares for and has responsibilities to his partner and children but I suspect that you would nitpick every one, because you aren't ready to accept it. Fundamentally you don't believe you deserve to be happy and safe and you refuse to believe that your children might notice or care that they live with someone who seems to be incapable of loving and caring for them properly.
I understand that the practicalities of being single are daunting and you are terrified of taking action because of the uncertainty incase it makes things worse. No matter how awful the current sitution is for you, you are confident that you are shielding your children from harm. But as defensive as you are on this subject at times, you have also expressed in your posts that you aren't confident that you can go on living like this, that its making you ill, that your bitterness is somewhat shared by the child he doesn't favour etc. I don't think you are able to keep this 'happy families' act up forever, even if you wanted to.
Please consider counselling or therapy for yourself in order to work on that self-worth, and get advice about the financial practicalities of a split so you can make informed decisions.
All this, he's not a (real?!?) alcoholic stuff... it sounds exactly like how the drinker themselves justifies it. If he isn't a real alcoholic, why is his er... hobby of drinking and smoking all evening more important to him and you than him helping around the house or being an effective father? Why does his choice to do this come above your right to be happy. If its not a choice, then he needs help, and if he won't seek it then he's a 'bad enough' alcoholic that its affecting you. He doesn't need to be falling down drunk for it to be bad enough to leave.
You may think he's functioning because he's holding down a job (for now), but he's not functioning after he gets home, is he?
Do you really expect us to say that its okay for you to exchange love, safety and happiness for a tiny bit of financial security (that isn't that secure, many employers will see an alcoholic unwilling to seek treatment) and familiarity? Its not something I'd want for any human being.
If/When you do leave him, you should probably spend some time recovering, going to Al-anon/counselling/freedom programme etc and just enjoying not having the weight of responsibility for his Alcoholism and finding your own, more reliable financial security. But you know, you might one day want to have a relationship with someone else who would do their fair share of the housework, cooking and childcare. Who would treat you with care and consideration. But you'd have to be free of this current (lack of) relationship first.