Hmm what do you mean you "allow" mil to watch her once a week? Mil is doing you a favor by providing free childcare. Your mom doesn't want to uproot and provide you with FT childcare, that's why she's all talk.
I pay for 5 days/week of daycare at the daycare center, regardless.
Whoa, that isn't the point at all. MIL is giving her time to take care of your child, and not charging you for it. Obviously you and your husband have determined that this is advantageous for your child, one day a week, or you would not be doing it. If this is your outlook, better to say no thanks and go with the daycare for five days a week.