*To spare her feelings dh and I have told her the nursery only had a full time spot for our child.
My mom also offered to do child care but to be fair we just decided to do full time care.*
You have TWO sets of grandparents - people who actually love your baby. That's something very precious. I'd think very carefully before handing him instead to people who are just paid to mind him. Who would you rather have looking after you - someone who loves you, or a paid stranger? Why would you pay to have your child cared for by people who don't love him?
Are you in the UK, OP? You wrote "mom" so I wondered if you were from the USA or another country or culture with something else going on that we're not getting?
None of us knows your full situation, OP. It's clear you do find your MIL annoying. But we're none of us perfect, and maybe there are ways you could get along better. Maybe she does need to calm down a bit, but the way you're talking about her makes me so, so sad, as a mother of several sons. Can you work with her, in your baby's interest? Eg you can say "I'm just not ready to leave him with anyone yet, and I know you love him, but this constant pushing makes me feel even more anxious about leaving him."
It sounds like there are some really harmless things that you are getting wound up about. Kissing your child on the lips isn't weird or gross - lots of families do that, whether babies or adult children. Baths for fun - tried and tested! You say your doc has advised every other day, but this isn't cast in stone and other docs might well advise different, eg if it's eczema, so MIL might just be asking or voicing another opinion. This makes me wonder if the other things you think are wrong about her are just as arbitrary. Maybe some are; would you at least consider few possibility that you could be mistaken about some of this?
You have the opportunity to increase the amount of love in your baby's life, by allowing one of the people most invested in him to look after him. Maybe not now, but please, when he's old enough - let him face a relationship with his gran separate from his relationship with you. Let him spend time alone with her.
My dad hated his MIL and would have quite liked to shut her out of my life when I was a kid. I'm so glad that he didn't manage to, because she was a wonderful figure in my life. Yes, she did some things differently from my parents and she was definitely wrong about nutrition and no doubt various other things, but she loved me, and I loved her. I think kids have a right to their grandparents, unless the grandparents are actively bad.