Maybe your MIL needs you to be more direct? You said you've run out of ways to say no nicely. Maybe she's one of these people who doesn't get hints and needs to be told things straight?
It sounds like you have allowed your MIL to pressure you into doing something with your child that you aren't comfortable with. It causes stress. The baby will pick up on this stress in his parents. You do not owe your MIL any particular grandma experience. She has had her turn with her children. Now it is your turn to make decisions about how you will raise your child. You can say no to anything at all. If MIL gets upset or reacts badly, that is for her to deal with. It doesn't mean you have done anything wrong.
A nine month old needs his parents more than anyone. Tell MIL, "Thank you for your kind offer. We don't need childcare at this time but will let you know if we would like your help." When it comes to overnights: "Thanks for the offer but we prefer to wait until he is older before we do something like that." Or, since you've already been doing them, "We have found it difficult to have our son away from us at such a young age and have decided that we will not do it again until he is much older." Any objections, just tell her it's what you've decided as his parents. She doesn't have to agree or like it.
MIL doesn't have to have your baby overnight or for childcare to feel included. She can visit during the day and read him books, help feed him if that is something you are comfortable with, take him for walks (with or without you, depending on how you feel about that), go on outings with you both, come for lunch in the weekend. She doesn't need to have alone time with him. If she is not happy to accept what is offered, this is not your problem and you don't have to correct it.
I'm sorry if this sounds harsh but I see a young mother who is doing things she is uncomfortable with to please her MIL when there is no need for you to cause yourself this stress. Your baby, your way. No-one else gets to make those decisions. Anyone who doesn't like what you decide, including grandma, will just have to get over it.
What do you think is best for your baby? That should inform your decision, not what is best for MIL.