As above, as long as you are entering as a British national on their Visa Waiver Program (VWP), i.e. on a tourist visa, which includes the ESTA, then you only need the passport to cover your intended period of stay. A return flight ticket will be enough if this is questioned at immigration.
Sometimes airlines are overly picky about passport dates.
Airlines are under strict rules to only fly people out if they have a good chance of being let in. It makes sense when the deportation/repatriation cost can be charged to the airline if they didn’t follow procedure.
If the airline question the duration of the passport, just politely ask them to “check TIMATIC”. It’s the IATA’s list of all visa requirements which the airlines stick to and it confirms the above.