IMO, you have the upper hand at the moment because supervised visitation was court ordered (right? Unless I read wrong). If that's the case, then he needs to petition the courts for unsupervised visitation and until that happens, you don't need to do anything continue on as you have been for the past 3 months.
He may send your attorney all the fun little AA reports he wants, but proof is in the pudding. IF you end up in court fighting against his petition for unsupervised visitation, THEN you have your attorney draft out a sworn affidavit detailing his alcohol abuse over the years. Perhaps start talking to people who know your ex to see if they would be willing to provide written (and notarized) testimony about your AH and his alcohol abuse.
In the meantime, I applaud you for thinking of your children's safety before anything else.
And FTR, I went to court with only my word about XAH's alcoholism and abuse. He knew when and where the hearing was to take place. He didn't show and then it was game over for him. Don't lose hope. Things CAN work out...trust in HP.