Am I correct in thinking that you need a minimum of 6 months? Just been doing that ESTA thing and noticed that my missus passport runs out on 22 june and we are going from 3rd Jan to 23rd Jan.

Hope the passport office is open in Peterbrough this week.

That's right 6 months minimum.

The system should reject the application if it's invalid you would hope, be careful with the immigration control people as well they can be difficult and pull you up on stuff like that if the mood takes them

That's not quite true. Check out the information on this link, http://london.usembassy.gov/faq-passports.html