You CANNOT cross babusar pass on a city. The road is too bumpy, loads of water crossing, glacier crossings,etc.
Its not just that you will damage your car but you will get stuck as some parts are only jeepable due to deep cuts from glaciers and flowing water. May be in a year's (or two) time when FWO can finish the metaled road it might be possible.
Also the road is too steep and rather difficult that driving should better be left for the locals. However, if you think you're not afraid of heights and are experienced driver (in mountains), you might want to hire some jeep for your trip. Even then, try to get one with a short distance between front and rear wheels (cant recall the technical term for this). I have seen land cruisers and hilux getting stuck en route to saiful malook by the inexperienced drivers.