I'm not sure about UDK but XNA for example can load multiple 3dsmax files and play the animation step by step.
I would create 1 3dsmax file with the character shooting the bow,
with frame 0 as default pose, frame 1 to 500 (for example) the animation of grabbing an arrow,
frame 501 to 1000 to shoot. (frame 1000 is the character in rest stance, like frame 1)
Then I would create a new 3dsmax file and merge this old file, delete the character from it leaving the bow untouched.
Next I would create a second file and do the same, but delete the bow and arrow.
Next step is to import the two files into your game, set the coordinates correctly so the character and the bow are aligned correctly.
When the user shoots just start playing frame 2 - 1000
The arrow is a different story, when it leaves the bow, all kinds of physics are applied.
I would create a new model for this and insert it into your scene as a separate object,
on which you can apply game logics (collision detection, physics etc)
For examples on creating animations in 3ds max I suggest a google search.