If you use a map case like an Ortlieb you can stop almost all of the flapping by tying a loop in the cord to shorten it. Now attach an Alpkit mini karabiner and clip it to one of the loops/straps on your rucksack. On my sacks I can always find a strap that I can reach easily without having to take the sack off.
A pack weight of over 20kg sounds very heavy. Did you carry the dog in it You should easily be able to get it down to around 12kg, even with quite a heavy tent. Carrying a lot of weight makes a huge difference to your speed and perceived amount of effort climbing up hill.