Do you know any veterans who are sober alcoholics? Or call the VA and ask someone there for assistance. Your marine will listen to another marine who has had a drinking problem and done something about it. You've already told him how his drinking is affecting you and the family. Now is the time while the relationship is still relatively new to get things sorted out. If he can control his drinking, great, if not he has to quit or somehow not bring his drinking problems home to you and the family. Be as non confrontational as possible but be firm. Dont call him an alcoholic, let him decide that. Let him know that you care for him but not his drinking behaviour. If you have talked to someone who knows about problem drinking and they are willing to talk to your SO then give your bf his number.
I knew an old married couple and the husband had once had a drinking problem. Actually it got pretty bad, alcohoic bad. Well one day after who knows how long the wife finally said to him, "You can stay, but that **** (alcohol) has to go." And that was that, He was not as bad an alcoholic as he might have been and so he was able to stop drinking. It scared him enough so that he quit cold turkey. I heard this story years later when they could both laugh about it.
This approach worked for one couple and might work for you, but it might not. How much crap are you willing to take?