Since you have an account now (London) and aren't looking to open an offshore account for illegal purposes, why don't you sit down with an officer of the bank and explain what you want to do?

If I'm understanding correctly, you want the funds (USD) available when you're in the US. You'd like access to them without paying exchange fees. If the London bank values your account I'd think they'd be interested in helping you. Maybe there's a way around the income verification. They very well may have a working relationship with one of the offshore banks.

Of course it may be totally different today, but the exchange fees banks used to charge were strictly for handling. They weren't a source of profit. They offered the service because their customers needed it. If that's still true, that should work in your favour.