Essentially you want to exercise the privileges of a rating that you hold i.e. for FAA IR. That should not pose a problem.
Secondly you are legally entitled to fly the aircraft in UK (International) airspoace both IFR and VFR so don't see why the VFR only bit comes into it. The instructon is for a FAA rating, therefore nothing to do with UK ANO as its not for the issue of a UK licence or rating so you don't need UK FI rating.
All seems quite reasonable.
Next question is one of payment. If no payments were made, very benevolent of you to provide free IR training, it would be a private flight and quite legal. To hire the aircraft to the student it must be maintained to public transport standards, not private.
Therefore it would seem possible to do it for FAA IR but you could not do it for a JAA IR.