Nobody likes to admit it but our income level impacts how we perceive our identities. I imagine BM in part wants to offer her daughter these experiences because they will be positive for the kid - but also because she likes knowing that her daughter "has it as good" as she did. Even if that's true, it doesn't change the fact that these are things the kid wants to do and could benefit from. So it's okay to let her.

It seems like your husband is caught up in his own identity. It's sad that he has been forced to tell his daughter 'you can't go because I don't want you to' and that didn't wake him up enough to make him realize he's holding her back to avoid feeling inadequate himself. He should be able to put his feelings aside and let her enjoy things unless he has some motivating reason to deny her from the schools, camps, etc that isn't mentioned in your post here.

The reality is, she has grown up with the lifestyle that her mom has provided. She's going to perceive herself as being part of the rich kids group. If your husband can't get past his feelings of being put-off by her mother's lifestyle and rich people in general then his daughter is going to end up feeling like he is prejudice against her because she IS part of that social group. He needs to deal with his struggles to accept the income gap or lifestyle differences so that it doesn't create a wedge between him and his daughter.