we 5 friends did same route on civic 99 a year back ....
no issue for 06 city. as it has good clearance ....
its kinda steep from astore city to rama ptdc and leave astore city for chilum chokki ...
but do able if once u reach astore, drop you luggage at hotel and keep the car with minimum load.
there are some sharp potholes before reaching astore from kkh, one should drive slow on those and pass them with care, as we burst our car 2 tires at the same time on those potholes due to car coming in front on a sharp turn, but people of that area are so nice, very helpful that even this happened in late hours like 8 pm (which is late in that area ) they traced the person and open the tyre shop for us, give us 15 km free ride to shop and back to our car .... much obliged we are .....
Astore is little dusty town, so keep in mind .... it has fuel, cellular networks of few, many shops ....
Astore to rama is quite steep, for city 06, one need to keep in little high raves but its narrow and steep, so just be careful ....
Astore to chilum chokki not an issue at all ...
after chilum choki , its rough track, and steep, u might face difficulty while ascending last few turns to sheosar lake, but its doable as u guys aren't car full of load.
i suggest, one should have good hard tires for such trip, like of bridgstones, general etc, keep tire pressure solid 32 minimum. spare tire should also be in good shape, all tubeless, air compressor, puncture kit.
there are few stream crossings where they were building bridge last year from astore to deosai, i have no idea if they are completed or not, if not, at those points i would suggest, drive very carefully as front bumper and silencer might touches stones there ..... otherwise no issue .... all the beating is for tires and at few places for engine (just before sheosar and from astore to rama) ....
have fun !