Your fundamental mistake was to book on Delta, as they will exploit any opportunity to make money out of your circumstances.
I would contact the UK Border Agency and ask them for some kind of letter which you can give to Delta. The truth is though, for Delta this is not about "helping you through immigration". Delta staff are trained to look for any reason whatsoever for them to grab "ancilliary" income from their passengers and the company has specific targets to generate this income. They are like Ryanair, and any excuse to milk you will be exploited. I doubt very much they will refund you no matter what you provide.
I would seek something from the UK Border Agency which states that you would not have been denied entry to the UK and that the airline was wrong to deny you a ticket on that date. If you can get that, you could then try complaining to the DOT.
Ultimately, the lesson here is for international flights, NEVER fly Delta. They are horrendous.
Link to UK Border Agency here:
http://www.ukba.homeoffice.gov.uk/partnersandfamilies/