If the contractor is paying for the hotel, why is he not paying for your meals as well? I was working for NATS on an airspace trial a couple of years ago, as they needed a military ATCO, and they picked up the whole ish.
If the RAF was picking up the tab for the hotel, I would expect you to have been told to stay at the nearest Service establishment (distance from work/travelling time permitting) and mess there for breakfast and dinner and claim actuals for lunch. You state the the nearest camp is 12 miles away - a reasonable travelling time/distance at either end of the day but it would not be cost effective to make you do a 24 mile round trip for lunch.
Ultimately, costs lie where they fall, ie your unit pays for what you do if the contractor is not picking up your subsistence as well as the accomodation. You need sort this out with your unit budget manager before you claim, as he will need to authorise anything you claim on JPA.