You may well be right A and C but that firm runs under an AOC. It IS possible for an AOC holder to get DfT permission to use foreign reg aircraft, subject to commercial need and subject to their UK competitors not objecting.
I read the above, pretty clearly stated, on a CAA or DfT website when researching the whole business of rental / dry leases (the subject of many megabytes of disinformation in various places) as applicable to N-reg planes in private use. Eventually I phoned up Roger Kinsey at the DfT who confirmed that for private flying it's all a redherring and there is no problem. Obviously I did not pursue the AOC side because it doesn't apply to me. Since then I have not found anything disproving the foregoing; however I am taking about noncommercial use whereas you are looking at flight training.
It could be that some more recent EU law supercedes Article 140, however. In which case not many people know about it!