How to tell a cousin they aren't invited

I had to do the same, I let them know I loved them but I hoped they would understand that I was not inviting them. We are adults and there's no need to feel obligated to send an invitation...
That was all I said. They understood.
Good luck, it's you're wedding, you should have unnecessary drama!!
Enjoy and congratulations!!

I wouldn't say anything unless asked specifically. And I'd just be upfront that you're uncomfortable with them for whatever it is they did in the past that made you feel like they wouldn't behave at your wedding.
I would avoid mentioning alcoholism or mental health as the reasons because you will sound like a huge jerk (and as someone with family members with alcoholism, heroin usage, and mental health issues all over I think you do sound a little harsh but not out of reason).
It's always best to point to a specific series of behaviours and say they did not behave right at X event in the passed so I am not comfortable with risking that behavior at my event. Any push back mention you are not close and have not seen any improvement so you are not taking the risk. You'll probably have to put your foot down and risk having the aunt and uncle not come.

Are these cousins people you rarely invite to your home, or people you invite over regularly, who invite you to theirs, sharing meals, special events and such?
If you do have each other over and do things together as part of your regular lives, plus holidays and special things, it will be hard to leave them out as far as other family are concerned.
But if you routinely dislike the drinking or boorish behavior , or just are not close, and usually only see each other at other people's special occasions, and there has been little reciprocal entertain for a few years, doing things together or at each other's homes, they you have no reason to invite them to begin with. Invite the aunt, uncle, and cousins you regularly enjoy seeing. They are close to you. And do not invite these badly behaved people you do not feel close to to begin with. This is an easier position to defend.
When people advise you to invite people in circles by closeness, it does not mean by branches on the family tree. It does not mean if you invite one aunt or uncle you must invite all. Or if you invite one cousin, you must invite all. You can invite all the friends or extended family you like, feel close to, and socialize with on a regular basis. And not invite those in the next circle out, casual friends, coworkers, or extended family whom you rarely invite to your home or do things with in a regular basis, even though you may see them daily at work, in a common activity, or at mutual friends and relatives homes, but never invite to each other's lives on a regular basis.
And all you have to say about this is, we invited aunts, uncles and cousins we socialize with regularly, and feel close to. And did not invite family or friends we rarely see outside of big occasions.
That is acceptable etiquette for any party or celebration, or milestone occurrence like a wedding. It would be wrong to exclude very close friends you spend a lot of time with regularly, in favor of extended family you spend as little time as possible with. And you can only have so many guests.