You can easily travel on KKH on your bikes during June. However, if you want to avoid bike travel but still want the luxury of bike after reaching Gilgit or Skardu then it is also possible. You can take your bike on the very same bus you are travelling. When I was in university, I had many class fellows from Gilgit-Baltistan and during long semester breaks, they took their bikes with them to Gilgit or Skardu and bring these back when vacations were over. I am not sure about NATCO but other bus services like Mashabrum travels do take bikes on bus roof. I myself saw bikes loading on Mashabrum bus when I was going to Gilgit.
Regarding security threats, just avoid to travel during night especially the Kohistan section on KKH. Govt also provides police escort convoy from Besham to Chilas, you can also avail it. If you are on bus, they will manage it all.
If you want to make real use of bikes, I'd recommend to travel from Rawalpindi on your bikes as the journey would be an adventure itself.
If you have anymore questions feel free to ask 
