Throw the handle underhand, quickly and accurately. Aim it in the direction of it, and start low so it'll hit its face or upper body. You should hopefully distract it.
Then, while it is distracted for a split second, run up and do a flick motion to chuck some acid at it (the stuff in the flask). Use the time to run past, and pour a bit more acid as you run past it. Then, run past fast. Hopefully it'll be on the pathway to the hawk's cage if it follows you.
This is why as you run past, you leave the bow on the floor to hope it trips on it.
There, done.
EDIT: An underhand flick, something like that.