Hmm yes,this is the section that concerns me the most;Diyarbakır has always been a volatile place and is now like a tinder box,best avoided as much as possible.
From Kayseri to Malatya there are no problems and from there you could drop down into Adıyaman to visit Nemrut Mountain(where there big stone heads are),you can go up there with no problems before winter kicks in.
After that you could take the ferry across Ataturk Dam and go to Şanlıurfa which,in spite of it's southern location is safe enough and a truly lovely city to visit.Up to this point I feel confident you will encounter no issues at all other than too many people wanting to offer you food and accomodation :)
Then? ....well you could cycle along the southern shore of Ataturk Dam through Siverek to Diyarbakır but from there you have some real bandit country ahead of you and it is at this point when taking a bus to Van would be a smart idea IMHO.
An alternative is to go from Urfa to Mardin but this takes you along a very major road which is always packed with oil tankers and large wagons plus military vehicles.Not ideal for you.But...here's an idea.You could take the bus from Urfa to Batman(5 hours very regular buses)and go to visit beautiful ancient Hasankeyf which will be lost under the waters of a new dam next year.Spend a day or two cycling around the area and then board another bus on to Van,these too are very regular.
This second option would be a good one I reckon.This option would allow you to visit some special points of interest safely and the go on to the next one safely too and avoids you having to cycle across sections where you would be vulnerable.
I know it maybe feels like you'd be cheating a bit but frankly who cares so long as you can live to tell the tale of where you DID cycle?
So that gives you a few options to think about but before deciding wait until you are in that locale,talk to people who know what is happening(those in authority)and then if you don't feel you want to risk it at least you know you have the buses there as a plan B.
Of course there is the perfectly possible chance you could cycle the whole way without the slightest problem but my conscience says I have to be honest with you and I make no apologies if it seems like I'm fussing.