Any car, if it is fit, can go to Gilgit. No problem at all. (Even you can go to Khanjrab Pass in your car.)
No experience of a tour during Ramazan. But I travelled from Lahore to Skardu during Ramazan in 2012 and faced no problem.
If you opt Besham-Chillas route, the road from Pattan to Dassu (36 km) is in a bad shape. Onward up to the Raikot Bridge, the road is average. Anyway, it is manageable, you just have to be extra careful. The road beyond the Raikot Bridge is excellent.
If you choose Naran-Babusar-Chillas route, hopefully you will find it open in first week of July. But you will have to make sure about the condition of the road.
Normally, petrol is not a problem. If you go through Naran-Babusar route, fill your car tank in Mansehra. (You will find petrol pump in Balakot, too). If you go through Besham side, fill your car tank in Besham. Again fill your car tank When you reach Chillas. (I have travelled between Lahore to Skardu more than two dozen times in last three years and have visited other parts of Gilgit-Baltistan too, but never faced any problem regarding petrol.)
You can make this tour in five days, but it will be a very hectic tour.
From Lahore to Gilgit, it takes almost 20 hours in one go (Chillas-Besham route). I did this in March this year. Left Gilgit at around 5:15 am and reached Lahore at 1:10 am.
Better to read some threads about the places you want to visit.
