Avoid night driving from abbotabad onward. Lahore to Naran in one day is possible but it depends on the route and the fact you had enough sleep before you leave. If there are more than one drivers than no issues. Take the motorway all the way to abbotabad.
I suggest you leave at midnight and try to cross abbotabad before 7am and you'll be in naran around mid day. If your drive through the night and get to naran in the morning, you'll be too tired to go any where and enjoy, and probably end up sleeping all day.
Take breaks to stretch your legs, don't drive fast and try to rush, it only saves you few mins in the end at the cost of risking many lives. Remember, it's not a race, take your time, enjoy the scenery.
July will be expensive. I usually travel in april/may or aug/sep, not to save money really but because it's quieter and calmer in those days, money saved is just a perk. I don't like rush and crowd.
I am planning a similar trip but I will go for at least 2 weeks in mid of August.