A UK examiner, holding both EASA and UK privileges, cannot examine a UK pilot in EASA in a EASA aircraft without approval from the EU state (airspace) where the flight takes place (according to the UK CAA).
Sadly once again the CAA are giving incorrect advice.
Examiner holding an EASA licence doing something in an EASA aircraft in EASA airspace - no permission required from the EASA state. They are operating within the privileges of their licence, and there is no issue with the aircraft being used as long as all the normal NCO/NCC requirements are being followed as if it were an EASA training or checking flight being conducted.
If the aircraft is UK registered, (and UK operated) then the examiner has no need for an EASA licence in EASA airspace, but in principle needs permission from the state for the flight. However many states give an exemption for training even when paid for. This has to be checked on a state by state basis. I'm guessing from BH's location that Spain requires it, but Germany, Belgium and Denmark don't. Not checked any others!