I don't have first hand experience of this situation but I expect that a print out of the ESTA website showing it's validity and number should be enough for BA to sort this out (perhaps at the Visa Check desks next to check-in). When printing many browsers often automatically include the date you printed in the header or footer; if it's not present, it might be worthwhile selecting the option to add as it give you another minor point to prove your case.

Good luck and do let us know how it goes.