Advice needed🤔 (MIL drama)

So long story short my MIL is unmarried and has a new boyfriend in her life about every year or so and the man she’s with now and has been with a little over a year is quite a bit older than her and an avid drinker. He’s nice and has always been very welcoming of us into his life. My place of concern is I’m going to be having my daughter any time in the next month and I do not want him to come to the hospital. I am already worried about having my immediate family visit in the first 48 hours of her new life and the thought of her being passed around from person to person is driving me insane. Although I love my MIL and she’s always been wonderful to me, she is so very pushy of her new men in her life on my husband an I. Always bringing him to every event (she even tried to bring him to my baby shower after I had already expressed to her that is was girls only) I get she wants everyone to like him but in the four years my husband and I have been together she has had four boyfriends none of which last and I don’t feel the need to accommodate each and everyone unless Ik they are going to be around for a long period of time. So in short, how do I tell my MIL she is more than welcome to be at the hospital during labor and delivery as well as after but I don’t want her boyfriend there as I am not comfortable with it, and do not know him well enough to want him to be there holding my daughter in her first few days of life. He’s more than welcome to come visit when we’re home and settled and comfortable being new parents. I know this was extremely drawn out but I’m running out of time and just don’t want to cause a big argument as she likes to take everything to heart.