J.S.
Your daughter won't get used to your MIL unless you allow them to spend more time together, which your original plan would have allowed. You can stick to the original plan of letting her babysit AND spend more time with her in addition to that.
I don't believe you should be uncomfortable with your MIL for having said that she didn't think your daughter was crying "that loud" because I believe that she was trying to be reassuring. She probably felt awful about her own granddaughter not knowing her well enough to be soothed in MIL's own home, and was also trying to reassure herself that the situation was not really that bad.
And as much as your daughter may have been crying, the crying in and of itself isn't bad. Your MIL wasn't abusing her or making her cry. Crying for a few minutes because she missed Mommy and it was in an evening where she was probably tired and wanted to be in bed doesn't mean you should take it out on your MIL.
So yes, your husband is right. You won't be "sacrificing your daughter's happiness." That's such overly dramatic language that isn't true. Your MIL isn't going to scar your baby. You need to let go a bit and trust her. Your daughter will read your cues and follow suit.
EDIT: Also please remember that your husband has a say in this. You don't want to make such a huge stinking deal over something that is so very minor that you cause a family rift. The grandparent relationship is a valuable one even if you don't value your husband's mother. The woman is not going to harm her grandchild, and in fact if anyone loves that baby as much as you do if not more, it will be her grandmothers.