The best and easiest time to change your name is just before you apply for a new visa (within 3 months of when you plan to apply). As you have already applied for FLR(M), and you still have 4 years left on your US passport, I would leave it for now, and then change your name in your passport a month or two before you apply for ILR.
The name on your passport and the name on your visa must match exactly at all times, therefore, in order change your name on your visa, you need to:
- apply for a new US passport in the new name
- apply to change your name on your BRP card... which costs £161, involves submitting an application and attending a UKVCAS biometrics appointment, and it can take 8 weeks to be processed.
You must change your name on your BRP within 3 months of changing your name in your passport, or you can be fined £1,000 and have your visa curtailed or shortened.
However, if you wait until you're within 3 months of applying for your next visa, you can simply:
- renew your passport in the new name
- apply for the next visa in that name within the allowed 3 months
... which means you do not need to pay the £161 to change the name on your BRP.