We are leaving for chilas on sunday, will update on the situation of babusar. I would suggest keeping snow chains as a precaution. I am also trying to get some made locally before leaving.
If the road is clear just take it slow, i have seen people absolutely flooring when going uphill and be on the brakes constantly when going downhill. I have an automatic, i let the car go in L gear and brake with gaps, that way the brakes have a chance of cooling down slightly. Also don't try to cool down the brakes by throwing cold water. Last time i was there some guys in a camry flew past us and were stopped further down hosing down the brakes with cold water. It can warp the discs. Keep driving slowly to cool them instead. Just use first gear going up and down in clear conditions. Make sure brake pads/discs are perfect. Also the cooling system should be good with proper coolant not tap water.
If it's snowing be ten times more cautious, brake and accelerate with a really light foot. I noticed that fresh snow has alot of traction to offer as opposed to frozen up snow where the wheels keep spinning no matter how light you are with the throttle and brakes.
In Nathiagali coming downhill in fresh falling snow i had to keep the brakes pressed nearly all the way as i didn't wanted to gather speed in snow with no chains and summer tyres. Brakes were definitely starting to heat up. Babusar is alot steeper, i wouldn't suggest going downhill in snow without chains. Although winter tyres would be perfect if you can find them.