Any car - and I mean any car - from a Khota Rehri to a Rolls Royce, can go to Khunjerab Pass. Just make sure that the Khota is strong and well maintained.
Really, while the altitude is high, the actual climb is gentle. In fact, climbing back from Chilas to Babusar would be a bigger problem if horsepower is low and car is loaded. The rest is all about comfort and driving pleasure but any car can go. A powerful car can go more quickly, more comfortably, and with more load as compared to an old or overloaded or underpowered car, but they can all go...except for supercars such as Ferraris and Lambos which are the only ones that will struggle due to very low ground clearance.
Advised, however, is SUVs and Corollas. City is also fine but Civic must be carefully driven due to some bad patches from landslide that may cause the underside to hit. These are mostly from Kaghan till Babusar and from Chilas till Raikot bridge.
