The stress ECG is all about the Bruce Protocol which doesn't hurt any more than any other. You'll probably have to fund it yourself and if the treadmill test is what this is all about, make sure that you take the original trace with you and get the specialist to send you the necessary letter by email for forwarding to the CAA.
Spire hospitals appear to have cardiologists who practice the arts of private practice. The ones I've met have all seemed perfectly au fait with the style of CAA requirements and their input has always been accepted by the CAA.