I would like to add a little more information for my Brother from Karachi.
Keeping the fog condition that start from midplane i.e rahim yar khan to lahore to bhera
The suggested plan would be like
1) Get your rest at night (always).
2) try to keep you car tank full and keep food for at least 12 hour (snacks / pizza etc )
3) Plan it as a trip with due stops i.e Multan , lahore , khewra mines , rotas . Dunga gali/ nathia gali .etc you get my point.
Note : Haven't travel in Sind so you guys need to update on this for places, hotel and stops.
4) Travel by motorways where possible (Cost more but reduce wear and tear -> your and car)
A trip would be like
Move early from karachi like 5 or 6 am .... A stop at Rahim yar khan would due for night rest unless you can make it to multan (highly unlikely as single driver)
next morning leave to multan for sight seeing but make sure to leave by 2 pm to avoid fog in the evening . Hoping your stop would be lahore.
Once in lahore there are lot of thing you can see so it up to you. (Fog would be a travel issue from 8 am to 8 pm but it depend on rain and weather but you can alway moves inside lahore)
(Motorway M2 or GT road (bumpy cost less) your choice )Leaving for Islamabad would suggest a stop at khewra mines for the kid (on M2 ..3 hours stop) so better leave by 8 am from lahore. Once you are at kallar kehar the fog effect significantly reduces but at place it there.
From here you can either go to Islamabad and sight see faisal mosque .pir sohawa ..etc
Alternatively you can keep on going on M2 and connect to Hazara motorway and goto Abbotabad. to see snow you may need to go to nathia gali. (If it snow you need snow chain other wise don't go if there is snow on the road Very very dangerous).
and on on else need to create your return trip plan
So here are my 2 cents hope this help