But it seems to be a problem to cross the Bulgaria/Turkey border with a rental car. Does anyone have experience with that? Maybe a rental car company that allows border crossings to Turkey?
If you do a search with a car rental agency like Argus you will find quotes from a wide range of Bulgarian rental companies so you can then easily see the various "rules" applied by each of them (click on Ts + Cs). I think you will find that NONE of the "international" companies will allow transborder into Turkey. Even for some of the local companies, Turkey seems to be a no-no. I did find a couple however (search from Sofia Airport) -
Autojet - 65 euro extra
Top Rentacar - with BIG costs and restrictions - "Cross borders are allowed to Turkey, Greece, Romania, Macedonia, Serbia on request. Cross borders for the class MCAE are not allowed. There is an additional administrative and permission fees apply: 50 euro for economy vehicles (EDMR,HDMR, EWMR), 60 euro for compact, estates, intermediate vehicles (CFMR, IDMR, EDAR,EDAH) and 80 euro for all other vehicles (IDAR, FWAR, FWMR, FDAR, IVMR, FVMR, IFAR, FFAR). The mentioned fee is for 1 country only. For each additional country the fee is increased by 50%. The renter should provide us with a copy of his driving license and ID card by e-mail at least 3 business days prior arrival. The required security deposit is double."
We have done/investigated a lot of multi country car rentals in recent years (including most Balkan countries) and cross border restrictions or add-on costs seem to be becoming the norm. The name of the game is to make the headline cost as low as possible and charge extra for anything else. And cross border, especially to beyond the EU, adds a lot of risk - which is not insurable for a reasonable cost. Car rental companies from e.g Carcassonne wouldn't even let me take a car into Andorra recently without paying more!!!
If you only want to go to Edirne then look at driving to the border (Kapitan Andreevo) or nearby (Svilengrad), parking your car and taking a taxi to the border itself, walking across and then getting another taxi to Edirne - it is only c26kms. LP and other guides tell details - it appears that Dolmuses to the border from Edirne and vv are plentiful. You won't be the only person doing it and the Bulgarian "cost" of paying for a rentacar for a day to sit doing nothing is very little!. We were in Edirne itself 2 years ago - though my last experience on crossing this border was back in 1965!! You wouldn't need more than a "full" half a day in Edirne so you would have time to get to and from the Bulgarian border from elsewhere in Bulgaria, cross and back and see Edirne all in a day