In my case, it wasn't necessary to prove it. I just included it in my sworn affidavit that my XAH consummed 2-3 liters of beer on a daily basis, along with sleeping pills at night. I started documenting in a handwritten journal on a daily basis what he did, said and drank. I also gathered the receipts from all his alcohol purchases, organized by date. I used that to back up my affidavit. Seeing as my X never showed up in court to defend his rights, I never had to provide proof.
I think what helped my case is that I specifically mentioned that I did NOT want to deprive my daughter of her father's presence in her life, but that I insisted that she be cared for safely, hence my demand for custody. I think the courts will look more favourably on a parent who's trying to find a way for the child to see both parents than on a parent who's categorically refusing to let the other parent be in the child's life.
As mentioned though, your son is of age to testify as to where he wants to live. The mediator I consulted early on in the separation process also functioned as a "child rights' advocate", and he would often meet up with parents and teen/tweens to get a sense of where the child stood on matters of custody.
So, if you son doesn't want to be around his father, it would be entirely within his rights to refuse to visit. It might be hard to refuse, but just know that the option is there.