The RR Exe was an X24 and indeed flew in a Battle: but Fairey had their own engines, the Prince (H16) and Monarch (H24), both also in Battles, and both seemed to drive coax counter rotating propellers. The Monarch-engined Battle was sent to Wright Field for evaluation, and flew 250 hrs while there.
I agree, the photo couldn't be the Exe-engined Battle as that didn't have a contra-prop and it certainly didn't go to the USA.