Your mother in law is diagnosing without a license. chuckle. Her dentist hasn't seen the teeth and may agree that the orthodontist is doing the right thing. Or he may have another way to treat her. If your mother in law is willing to pay for a third opinion take her up on it. But if you're comfortable with your present decision and don't want to spend money on a third opinion don't take her to the dentist.
When I was growing up dentists did everything including braces. But I don't think I've heard of anyone having braces for baby teeth. Then again I'm not sure what your daughter's situation is.
If it were me I'd talk with my dentist, as you say you'll do. Your dentist will know if braces are ever an option in cases such as your daughters. That may help you decide whether or not to get a third opinion. It's my thought that your mother likes her dentist and doesn't know anything about what he'd do in the case of your daughter. Does she even understand what the problem is?
If you think it's appropriate to go to your mother's dentist for relationship reasons then you could have the orthodontist that you've chosen or your dentist to call your mil dentist first. The dentist may agree that braces are not appropriate. Your mil would be happy and your daughter wouldn't have to go.
Even if your mother in law were paying for all of the treatment you do not have to go with what she wants! Your daughter's health is your responsibility and not your mother's. My daughter frequently reminds me of that and she's right. I can make suggestions but the decision is hers.
If your mother in law volunteers to pay some of it but will only do it if you do what she wants I'd graciously decline her offer. You are responsible for her treatment. You would feel badly if the outcome was poor but you'll feel really bad if the outcome was poor and you allowed your mother in law to make the decision.