Thanks for the answer! It is a really strange situation.
So, in your example, the timing would look something like:
1. Attack (roll dice, resolve, count'em up)
2. Perform hazards in the chosen tree.
3. Trigger LH in the case where it's pushed.
4. Apply original damage
That could work, but would throw me for a fit if the LH stole the kill from me even though I hit her first lol.
This to me makes the most sense, it seems to be the most logical stack order.
Even the LH "Steals" the kill you still earn any favor to your pool the LH generates.