A week is too less for exploring Gilgit & Skardu. A better option would be to visit Fairy Meadows only which also falls in Gilgit-Baltistan region. Kaghan valley would be a good option too but it will be crowded like Murree during this season. No security issues in Fairy Meadows area and it is perfect place to visit with family.
First day, you can drive from Lahore to Abbottabad via Nathia Gali, can also explore a bit of Nathiagali if time allows. Second day, Abbottabad to Chilas via KKH, its a long long drive, night travel should be avoided so start very early from Abbottabad. Third day Chilas to Raikot, park your vehicle there and then take Jeep from Raikot to Tato and then trek from Tato to Fairy Meadows (Horses are available for people who don't enjoy trekking). Rent out a hut at Fairy Meadows Cottage or Raikot Sarai. On fourth day, you can enjoy the valley of Fairy Meadows, can trek to Beyal camp or simply stay at Fairy Meadows enjoying the sight of Nanga Parbat. By fifth day afternoon, you can start return journey, trek back to Tato, take jeep to Raikot and then reach Chilas, you can spend the night there. Sixth day you can reach Naran via Babusar Pass (follow this thread to know if Babusar Pass is open by then but most likely it will be open), Lake saif ul maluk can also be managed on same day. Stay night at Naran and leave for Lahore early on seventh day. You will make it by evening.
If you can add a day more at fairy meadows, making a total of 8 days then it would be even better.
Regarding your car and its capability, other brothers can comment better becoz I'm not experienced in this field 
