You should never attend a meeting sick.
Jehovah wants your best which means you should be genuinely happy and healthy and if you are ill you can't give your full attention, mind, body or strength to serve him. Please remind your daughter that she is showing great love for others by allowing herself time to get well. (Gently tossing the manipulation back in their own words.)
Her mom is very indoctrinated and thinks that the way she's taught at the hall is the best way. You know nothing could be further from the truth.
I wouldn't advise circumventing another parent's authority, but you can talk to your wife and help her find a way to give the baby an opportunity to attend without the Mom having to go back on her word and lose ground.
Even worse have you to allow the daughter to feel like she has power over both of you by overriding what your wife said. It will just cause hard feelings between you both and could quite possibly damage the relationship with your daughter in the future.
Your daughter is very intelligent.
With some thought and well placed words you can turn this around from a win-lose and make this a win-win scenario.
Hoping it turns out well.