Scorcher,
We drove over in April from S. Wales to Uzumlu with one dog.
Pm me for details but it was easy and in our opinion less stressful for the dog.
In brief our route was Eurotunnel to Calais, Calais to Switzerland, Switerland to Ancona in Italy this took us 2 and half days. Ancona to Cesme ferry 3 nights and then Cesme to Uzumlu. Great roads throughout.
You can drive totally overland but obviously that takes another 3 - 4 days.
Plenty of dog friendly hotels on route for overnight stops. Ferry runs once a week on a Sat evening getting into Cesme early Tues am.
I can be more detailed if you contact me through profile.
Cheers
Fi