What are you arguing about when he is drunk?
Why are you making him promise you that he won't drink?
Is she a drinker too?
If you make him promise not to drink, it places you in a position of equivalence with the drink. Its you or the drink. He will, if he is very badly addicted struggle to choose you. He will continue to choose the drink in the hope that you always back down. He will continue to choose the drink until you remove yourself, then once you have already gone, continue to choose the drink because its there. Its reliable and it doesn't walk out on him. At the moment he has a relationship with you and the drink. If you are not there, just as you were not there two years ago, he still had a relationship with it.
I posted a while back about DP and his drinking. Similar circumstances including the length of time we have been together. We have spoken about it, he has "promised to cut down" he has tried very hard. I never asked for any promises. I have not made my love or support contingent upon anything. I never will. Not because I will be forced to accept he might choose it over me, but because I know he has to stop, not for me, not for my approval, but for himself. He has cut back a lot, has spent more days without and than with and hasn't been drunk for weeks. Tonight he fancied a drink, apologised to me, then corrected himself and said "I promised myself, I'm failing myself" He had one drink before dinner.
But, he hasn't lied about anything. Nothing. and i trust him, this is partly because I have never put him in a position where he needs to lie.
However, your situation is different. For some reason your DP feels he needs to lie. You have assumed that if he needs to lie about drinking, this means he will lie about other stuff too. He might. because once someone feels they need to lie about one thing, and they do, they feel justified in telling lies about other stuff.
I would be interested to know whether this other woman is a drinker? A social drinker? someone who hangs out in the pub? Does he feel that this person would be less of a distraction and not stand in the way of his drinking?
But when all is said and done, if you feel that he must give up drinking for your relationship to survive, you'll be waiting a long time probably and it might be best to cut your losses now.