I am having the exact same dilemma!
I think the process is to first apply for the USA visa with name change. Then within 3 months we must notify the UK and pay a fee to get the new biometrics card. They said this can take up to 8 weeks. There's an application online to do this.
I think that's it but not sure!
I've noticed its also cheaper to change your name on FLRM than it is on ILR.
I'm wondering if there is a way to change our last names during the application process of ILR to save paying the fee to change names.