Welcome to SR Fees. You have found a great place with lots of Experience, Strength, and Hope (ES&H). Folks who have either been where you are now, or are at where you are now.
That being said, some of our responses may not be what you 'want' to hear, but they will be coming from our own experiences.
You ask what is the best way to be supportive? Work on you. Try some Alanon. Learn how to set boundaries and keep them. You will also find people that you are able to talk to and call when things get rough, and yes they may get rough. Alcoholism is a progressive affliction, thus his being a 'functional alcoholic' will change.
His 'agreeing' or 'seeing' that he has or may have a problem is just the first of many many teeny steps he will take. Or he may have just agreed with you (alkies are notorious for doing that, I know I sure was) to get you off his back. Only time will tell and since you believe there is a problem, there is a problem for you, it may not be one for him YET.
He will try and manipulate you, he may reach the point of blaming you. He could become verbally and physically abusive.
And then again none of the above may happen. However, please do not take his 'words' as gospel. His actions will be a truer exhibit of his commitment to recovery.
Sad to say he may have a long way to go before he really becomes "willing to go to any lengths to stay sober."
There are some excellent informative 'stickies' at the top of this forum, some more at the top of the '12 step forum' and even more at the top of the 'alcoholism' forum. They will all be very helpful to give you additional insight into what is about to happen in your life.
Being sober almost 27 years and being in Alanon almost 24 years, I have seen both sides of the coin and I can honestly say "it ain't pretty." However, for the non addicted Alanon can be an excellent support system.
J M H O
Please keep posting and let us know how you are doing, we do care very much.
Love and hugs,