I'm not entirely clear what you are doing - have paid for 1 F ticket, 2 Y tickets and then need the miles for a F award (doubled with the Amex voucher).

Assuming that is right, you could always ask to go overdrawn but I doubt you will have any luck unless you are a fairly high scoring Gold. I did ask once as Silver for a 2K overdraft and was politely told 'no'.

A couple of ideas:

1. Do a mileage run (the famous LON-BCN-AGP/SVQ and rtn). 2 of these would net you all the miles you need. Or you could combine one of these with an I class deal to Italy.
2. Transfer Diners Club points into BA miles
3. 5000 miles bonus for BA Amex - get someone else in the house to apply then make a household account
4. Spend £5K on the Amex card.