Get the £60-fee BMI Amex card, get your 24,000 mile bonus, get a further 14,000 miles for the £7k you are planning to spend, redeem your 38,000 miles for a J ticket to the US!
My BA Amex goes in a drawer when I hit £10k (actually, when I hit £9,800 to avoid early triggering!) and the BMI card comes out.