It sounds as though you are well, and that is the most important, everything else can wait. I'm so sorry about your chicken demolition crew. I have lost track of how many times that has happened to me. My hugel is in need of repair right now. I fenced mine and that took care of the problem, until I left the gate open one day.
As far as your dilemma, yes to the option that suits you best.
I personally would never deconstruct my hugel. I'm way to lazy, and it's way to much work. Between sinking from time and the chickens my hugel is much shorter then I want. I happen to have a bunch of tree trimmings. I'm going with a strange and unconventional repair. I'm going to pile those branches on the sides and on top, try to fill all the empty space with soil, then do a load of
wood chips and then, all of my chop and drop from my winter garden is going on, last I'm going to dump as much compost and soil on top as I can get my hands on. Then I'm going to plant a bunch of veggies on it and see what happens.
You may want to ask yourself, Do you want to start over? Do you have time to start over? Will you be happier starting over? If
the answer is yes, then you are made of sturdier stuff then me, and you
should start over. If your up for the work, and want more growing space and have the room for another hugel, then dump a bunch of soil, compost, what ever you have on top of your existing hugel and build a new one. If time or your health are an issue, you can do what you can for the hugel you have, use it and set the wood aside for when you have the time.
As for the tree if you are happy with where it is, leave it. If it will keep you from growing your veggies, move it. I would try to move it as soon as possible. Its best if you move it when it's dormant. Or the sooner the better. I always worry about moving a tree. I know it's done, its how we get bare
root trees, but its a gamble.
I'm sure what ever you decide will be great. Good luck to you.