In a pickle…

Devils advocate, you have a long life as her daughter and this sounds like a very minor speed bump of hurt feelings.

When she visits do you offer to let her feed and change baby or does she feel she’s overstepping if she tries to ask or help? The bath advice is not worrying tbh, we laid our babies in a shallow bath holding their heads to kick and enjoy the water. Clearly you might have differing styles but have you allowed her to help you or to see how helpful she is before you dismiss her?

Causing a rift not just between yourself and her is one thing but a rift will grow between her and her own son and grandchild if this grows. For the health of my marriage with my husband over a minor disagreement with his mum I wouldn’t push it and make amendments.

Stand your ground and she needs to respect your boundaries. But I'm with you. If she's not already making initiative during the visits to feed the baby and change her then how do we know that's not going to continue happening at her place who's going to change and feed her while she's at your mother-in-law's?

I would absolutely not be okay with him going over there without me. If she loves your child, then she can respect you, and acting like a child and going behind your back to your husband isn’t the way to get what you want!!

My child wouldnt be going anywhere unless im confident and secure with how she getting cared for.
As far as the relationship i agree with above comment. Us as mothers have to actively build our tribes and tribe for baby. Have u offered her to come over and been inviting like giving her baby and maybe just leaving for a few minutes. Try to be proactive about building a relationship because ultimately if anything does happen to u like illness or death…. More than likely she gonna be heavily involved so u might as well try to involve her so she knows and is aware and u can trust her. And u also will want your kid to have people outside of yourself they can trust. Research shows thats where kids thrive. I would absolutely bite the bullet and be proactive about building my babies and husbands support system! Later on she may be your future relief for babysitting etc so invest now!

My son is 16 months old and I’d only now start to feel mildly comfortable with my MIL keeping him by herself. She’s watched him for like 10-30 minutes with one of us not that far away. Hes stayed with my mom for up to 2 hours at a time but my mom is way more respectful of boundaries and understands current parenting much better. My MIL has also been treading right at the line of boundaries for his whole life. No one lives local so we don’t see anyone all that often.

Trust your gut. She can come to your house and earn those grandma privileges. No one has a right to your child