https://www.birdforum.net/showthread.php?t=266942&highlight=Hungary+for+More
Larry
Although it was back in 2013, I highlight a good traditional location for Sakers in Hungary. Unless the nest-boxes there have been moved, you have a good chance of connecting. I had 5 different birds when I went to that area near Hortobagy for a day. Little Hortobagy is also a good bet if the primary location fails you.
All 4 are possible in Bulgaria according to E-bird.org.
Levant Sparrowhawks pour through Burgas area late September, and also are breeding in south of Country when they can be seen in May.
You are right that SC Flycatchers can be seen close to the main road in Goritza Forest close to Burgas, but primarily in May and then early June.
Saker can be seen in Southern Bulgaria, but you will need precise information & most are the result of a reintroduction programme in 2015.
Likewise you will need precise info with Rock Partridge.
Bottom line is all 4 are quite difficult but possible. Though you could 0, 2, or all 4 depending on your information quality.