How to move a cat from turkey to the uk? (Istanbul)

I have recently decided to move my Turkish cat to the uk in the next couple of months.
I have read the paperwork from lefta and I have started the process of rabies shot and micro chip etc. so I'm 4 months she can move. But I'm having trouble with is how to book a flight with a cat with Turkish airlines (they are approved for travel to the uk I heard) and/ or a agency to deal with the whole move as I will be out of the country.

The only and cheapest way I can see is to fly through Istanbul pick her up from a suitable cattery (suggestions pls in the beyoglu area) and fly back with her as a carry on! Is this possible?