Assalam-o-Alaikum,
It is really nice to hear that you are taking your family to Naran. I am also planning to take my family, Wife + 3 kids (7y daughter, 5y son and 1.5y daughter) to Naran on 13th June. The time you mentioned is the Peakest out of the peak time given the conditions, circumstances and Ramadhan approaching. If you have finalized your trip, please go there with Allah's name and stay at burawai or batakundi. You may find some suitable places over there. In comparison with Naran, these places are much calmer and beautiful. I like calm places so I told you. If you can't stay over there, please consider these things while traveling to Naran.
Reach Naran Saturday Afternoon round about 12 PM or 1 PM. Reason: Almost 80% people tend to make exits from hotel rooms on Saturdays before 12 PM. Don't go Thursday / Friday. Monday morning is the best time to find some suitable rooms with in suitable fares.
Plan your trip and spend some time researching about the rates, fares and conditions prior to booking the hotel rooms. Since it is the peak season, hotel owners won't do much wear and tiers of the rooms.
Regards,
Junaid