As the Cambridge Tiger Moth School is a CAA approved training establishment, the problems with the QXC cannot be insurmountable. Oxford, Cambridge and Wycombe are examples of towered fields with grass runways (also Southend? can't remember) and presumably the school phones ahead to let them know that a prop swinger will be required. The recent article about the school in Flyer mentioned that the instrument bit of the course is done in a PA 28.
I'm not trying to underestimate the difficulty involved in doing a PPL in an open biplane, but it's not impossible. TM: don't be discouraged. Other options might include the Cub tailwheel ab initio course at Clacton. Good luck whatever you decide to do.