Why would you not just use your US Passport to check in when departing the UK and arriving in the US, and use the UK Passport when departing the US and arriving in the UK?
In practice this would work (and I've tried it) because airlines get fined if they don't check that the passport you present is valid for the destination, and if they are supposed to check that the departure passport is correct for the country you are leaving from then they don't seem too concerned.
A couple of years ago we spent 5 months in Australia and since it was cheaper and more convenient for a 6 month visa in our British passport this is what we did. At check in we presented our UK Passports since our US ones did not have a visa for Australia and again were never asked about there being no US visa in our British Passports (at first glance it looks like we had been in the USA illegally). Of course we had our US Passports on hand anyway if asked to present them.
Australia has proper exit controls so on leaving we checked in at the airline desk with our US Passports but at border control on the way to the gates we presented our British passports as that proved we had been staying legally in Australia. Like the USA, Australia requires its citizens to enter and leave on their Australian passports and our daughter's partner who is a dual US/AUS citizen does this every time he flies home. (AUS passport on the way out and and at AUS border control when leaving, but checks in with his US passport at the airline when leaving Australia).
