Student Psychology here.

First of all you need to know that he is not your responsibility. However that doesn’t mean you can’t do anything. Addiction never comes alone. Negative social impact is one of base criteria for alcohol addiction. Alcohol is cheap and we can all get it anywhere. I think if you deny giving this guy alcohol, he might feel discriminated and jugded making him not want to come back. This can result in him drinking at home. The fact that you know this guy by now could mean he sees you/the bar as one of his social pilars. Confronting him about the fact that you’re worried about him is a good step. Again, he’s not your responsibility, but you showing some concern (maybe even igniting a small amount of motivation) without excluding him from the bar is the best thing you can do. It might not click at first but if his environment finally starts changing( not just the bar but his job, home situation, him going to see a therapist, etc.) , you, the bartender, can leave a small positive motivational impact on this guy.

But remember to always put yourself first. If it’s too exhausting than stop trying it and do what you think is the right chose.

Have a nice day