For clarity, posters need to understand that in order to bring pets in, the aircraft and the operation have to be commercial, on an AOC, with approved paperwork for the operator, the specific route and of course the pet. In the past there might have been a few anomalies that slipped through the Animal Health vetting system but these days it's tighter and the forms to fill in very clearly and concisely identify which AOC the aircraft is on and that the aircraft is operated commercially. That means that say an IOM-registered aircraft offering to get your pets in will be doing so illegally - no IOM-registered aircraft are on AOCs.
As to the logic for only allowing commercial carriers to import pets, that's a separate debate, but those are the rules and they are very strictly enforced.
Talk to the pet-handling staff directly at the likes of Biggin, Oxford, Cambridge etc. who have built up a lot of experience and daily dialogue with Animal Health at Heathrow on these matters.