Bonk III appears to be based off Bonk's Adventure's engine instead of Revenge:
Address Table
I did a quick test TAS of 1-1:
Link to videoUser movie #52775436476338341
It's probably a bit suboptimal.
If you do like any hovering at all and then land on a slope it seems you land on your head like every time.
Personally I find this a bit less fun to TAS than both Adventure and Revenge so I probably won't work on this much more right away but I wanted a break from my current project so I took a quick look.
I decided to take another look at the second boss and found a trick that could also be applied to the first boss:
Link to video
So basically there's two parts to your attack when you attack in midair. When you first start to attack in midair it acts like a normal headbutt like while on the ground and attack to your side. Then you flip and attack below you and this does double damage. The trick is to hit with both parts of the attack and deal 3 damage instead of 2.
In order to do this you pretty much need to perform the attack at the corner of the bosses hitbox and this results in slightly slower attacking but saves time in the long run. Very precise to do though.
EDIT: Thankfully the game is hex friendly so I didn't have to redo any other stages. I will work some more on boss 2 though, that one is really tricky.