Turkish currently is the only one-stop, one-airline option allowing pets in the cabin on this route. All others either don't allow pets in the cabin (Cathay, Qatar, Emirates), have temporarily suspended carriage of in-cabin pets (Vietnam Airlines) or (temporarily) suspended the route altogether (Korean, JAL, Asiana, all pet-friendly European airlines...). Due to Turkish's unique policy of not allowing cats and dogs concurrently on any given flight, book your trip and have the animal confirmed ASAP with as much notice as possible in case someone with a dog also tries to be on any one of your desired flights.
Service dogs are the only animals allowed at the
Istanbul airport airside Yotel (and they are fussy enough about that...) so, no, breaking the journey there without Kitty having to enter Turkey is not really an option. I wouldn't even think about sneaking him/her in - as you are left with zero options if found out.Turkey implemented a two-week (in-home) quarantine requirement for all passengers with animals entering the country on May 1 last year, so unless you want your stopover last two weeks I`d suggest skipping it altogether.