In the cardboard box with the bike I also stuffed all metal parts such as tools, knives etc in the water bottles. I also put my roll with my tent as cushioning in the box to avoid a damaged bicycle at arrival. I think I also put my sleeping bag and mattress in the box. The trick is to put all the bulky stuff in the box and place it where it will cushion the bike. If needed put plastic bags around if your bike is dirty. Of course when doing this, observe the 50 pounds max limit.
I was able to put the remaining stuff into just 1 of my rear panniers which acted as my carry on luggage. Then a hip purse/bag for all my essentials and wearing my fleece/jacket if it should get cold during the flight. Wearing the fleece/jacket will again displace a lot of volume from your bags.