No, you will need to change your passport first.
This is because the name on your BRP must match the name on your passport at all times... and your new visa can ONLY be issued in the name on your passport, so it is not physically possible to have the BRP issued in your new name, but keep the passport in your old name.
The best and easiest time to change your name on your passport is to do it just before you apply for your new visa (within the 3 months before applying), so that your new BRP will be issued in your new name
The process to do that is:
- renew your passport through the US Embassy in London, which costs £91 and takes around 10 working days (
https://uk.usembassy.gov/u-s-citizen-services/u-s-passports/renew-an-adult-passport/)
- apply for FLR(M) in the new name on your new passport and your BRP will be issued in your new name
And that's it.
However, if you don't change your passport name now, the process will be:
- apply for FLR(M) in your old name (the current name on your passport) and your BRP will be issued in your old name
- change your name in your passport through the US Embassy for £91
- then you have 3 months to change your name on your BRP, or you can be fined £1000 and have your visa cancelled
- to change your name on your BRP, it costs about £160, requires that you send off your new BRP (and possibly your passport too) and it can take up to 6 months to be processed!
