First of all, a baby will always bond with primary care givers first and foremost regardless of baby classes, how you feed her, or how many people cuddle her. DD has been around lots of family caring for her and watching her overnight since she was a few weeks old and while she has a good bond with them, she's closer to us. It did not stop her from bonding. It did create a good bond with other family members, which is nice because we can go out and leave her with them knowing she's happy and having a little baby vacation herself.
Most classes for sign language start at about 6 months (and by then you might want a sanity break!). I personally think it's silly, you'll recognize the different types of cries to what they want. DD has a very distinct cry for food, sleep, protests, or cuddles. But hey, it's fun to do I suppose and if it's your thing go for it.
Something to think about though, you'll be bringing your baby into a new environment with all sorts of germs. Who knows how well their immune system will be doing. I'm not saying don't take your baby out ever, but ask anyone who has a child in nursery. When one child catches something, all of them do. So things like granny 1:1 time is great. Baby 1:12 different babies and mothers from 12 different environments, not so much. Really, the baby isn't going to get anything from being around other babies. Last time DD was around another baby she had no interest in him and he poked her cheeks until she cried. 
No, but really. Children don't need to be around other children to be happy. They need a stable carer to feel safe and nurtured to get the development they need. The baby class would be more for MIL's benefit, not your child's.
That all aside, the fact that MIL is possessive of the baby already is not a good sign. It's also a bad sign that DH doesn't have your back on this. If you want to do the baby classes, you do them. MIL is a grandparent, not a co-parent. This isn't her second chance at being a mom. I'd remind her that this is a baby who will have their own thoughts and feelings, not a fucking puppy. 