salam
firstly Chilas is on KKH.
secondly, you cant take the Naran Babusar route till at least start of June as it stays closed due to snowing.
MG HS...hmm...this car requires high octane if I am not wrong and that would be difficult to find in Gilgit Baltistan.
Anyway.. regarding which route you should take...obviously you will have to travel on the KKH. If you are traveling from Lahore then you will take the M2 first and then from Islamabad toll plaza you will enter the M1. Then you will take the Hazara Motorway and go uptill Thakot and from there onwards till Besham. I recommend that you stay there for the night. Then leave early the next day and you will easily reach Hunza before sunset.
There is ofcourse the dam construction site in Dassu and so at times they stop the traffic for a few hours while they are rock blasting. The road condition is fine and KKH remains open throughout the year.
stay options include
Besham
Jaglot
Gilgit
Hunza
Avoid Chillas as temperatures there are relatively hot.
let me know if you have any specific questions and i shall try and answer those.