
I'd always use the airline of the entry country - from experience it seems clearence is faster that way, and as far as I'm concerned, not being kept waiting around at the airport on arrival is paramount, especially with a young puppy. I have never shipped at under 4 months (and that was from Gatwick to Geneva). My UK - Canada imports (when we lived out there) were 5 month and 7 months (separate times). And I have always flown direct rather than having to switch/stop over.
I do agree that I'd always fly with the puppy in the hold - not only for the reason Brainless has given (heating), but because in a flight of that length, it's entirely possible the dog may need to 'go' which indeed wouldn't be pleasant for fellow passengers!! In the hold, it's comparatively quiet, dark and with a regulated temperature - and the pup would be able to be in a bigger crate (within the requirements of the airline because actually the crate should 'fit', but not be overly big (turbulance injury). My Basset puppy who I accompanied (same flight) from Gatwick to Geneva was in the hold even (he was significantly large, even at 4 months however!).
ps Those we took or imported from the UK into Canada didn't have their rabies shots before entry - just a Health Certificate and the KC Export Pedigree so they could be registered with the CKC. I felt the trauma involved with the flight was enough, without needing to be rabies vaccinated which Canada at that time, didn't require in any case. It might be different with USA although I don't see why, given UK is rabies free.