Get a Diners Club card and hope that they have a double mile BA promotion this summer. If they have the promotion you may be able to get enough BA miles via signup bonus, usage of the card, and transfer of AA and UA miles to BA via Diners Club points (and in the meantime try to get your AA balance to 30,000 miles and your UA balance to 10,000 miles as the transfer from airlines to Diners must be in multiples of 10,000 miles).