IO540 sent me a PM with contact details for a local pilot friend of his shortly after my initial post. The local pilot was very helpful and made arrangements so that someone came out to meet us on the apron with the avgas fuel truck around 9 AM on the day of our departure. The price was 4 USD per litre, payable in cash in either USD or EUR.
Contacting the club through the hotel staff also resulted in a promise to get fuel in the end, but the outcome certainly felt more assured going through a personal contact.
Reading the relevant sections of the Turkish AIP prior to the trip, I could not find any mention of a need for PPR for EU-registered airplanes, and none was asked of us.
We paid around 90 EUR in airport fees (approach, landing, parking, follow me-car and marshall) and 318 EUR in handling fees.
There are at least two different handling companies at LTFJ. Make sure you know which one you have used when you come back to the GA terminal. We didn’t and there were apparently no records of our flight in the systems, which led to some unnecessary lost time until the right handler came to take care of us. Even then things were slow moving. We arrived at the terminal at 07:30 AM with refuelling planned for 09:00 and an expected departure time of around 09:40 AM, and we ended up taking off at 11:30 AM.
The other airport we landed at in Turkey (LTAZ Nevshehir), on the other hand, was almost as efficient as any Western European one, and very affordable at 60 EUR in total fees (no handling).