Hi Pete & Jo
Yes you can bring a pet over without first getting residency.
You need to make sure the airline can carry the dog, often more a physical space thing than anything else, and you need an approved dog carrier basket thingy - the airline can advise where to get these. Turkish Airlines seem to be the most receptive, Cyprus Turkish can do it on certain flights but if your dog plus weight of basket are over 32kg then they use normal baggage rules and may refuse to carry it.
As an extra piece of information, airlines nowadays will not accept a dog that has been sedated in any way, so be careful.
Hope that helps,
Scunner