Training for class rating revalidation, such as the one hour with an instructor, does not require a RF/ATO. As explained below, no additional permission is required.
You would need permission from the CAA to use such an aircraft, which could be granted to the owner for a fee.
Where the DfT issued ANO 2009 Article 225 permissions free of charge, this was handed to the UK CAA some time ago. They began to charge around £65 for each permission, where each permission issued was valid until the aircraft insurance or C of A was due for renewal which meant it was often only valid for a few months.
During the ANO Review (CAP 1414, CAP 1335), this was an issue raised by a number of stakeholders. ANO 2009 Article 225 refers to aerial work, which included a flight on which the flight instructor or examiner is remunerated.
ANO 2016 has aligned with EASA definitions and now talks of a "commercial operation", which would not include such a flight. Flight instructing and examining in EASA regulations fall within non-commercial operations.
ANO 2016 Article 252 now states:
Restriction on commercial operations in aircraft registered elsewhere than in an EEA state
252.—(1) Subject to paragraph (2), an aircraft registered elsewhere than in the United Kingdom must not fly over the United Kingdom for the purpose of commercial operations unless—
(a)the CAA has granted permission to do so to the operator or charterer of the aircraft; and
(b)any conditions, to which such permission may be subject, are satisfied.
...
Meaning of “commercial operation”
7. For the purposes of this Order, “commercial operation” means any operation of an aircraft other than for public transport—
(a)which is available to the public; or
(b)which, when not made available to the public, is performed under a contract between an operator and a customer, where the latter has no control over the operator, in return for remuneration or other valuable consideration.
Unfortunately, both the CAA website and Flight Examiner's Handbook are incorrect in their interpretation of the current regulations. Remunerated training and testing of an owner or part-owner in an aircraft registered elsewhere than in an EEA state (such as an N-reg) does not fall within the restrictions contained in ANO 2016 Article 252, and therefore no longer requires a specific permission from the DfT or CAA.
ANO 2016 states that the licensing requirements of the state of registration must be met. In the case of instructing and examining in a N-registered aircraft, the holder of a UK CAA-issued flight crew licence and instructor/examiner certificate are rendered valid in UK airspace by FAA regulation 14 CFR 61.3 and ANO Article 148/149.
The instructor/examiner should familiarise themselves with FAA regulations beforehand, and, if in doubt contact a FAA instructor for guidance. As mentioned by Whopity, in order to use the aircraft for a course towards issue/renewal of a licence/rating/certificate, the RF/ATO would need to check that the aircraft is suitable for use on that course and meet any Operations Manual requirements. No CAA involvement required.
There is, however, an issue with solo flying required on the course...
ifitaint...