You are removing the bark. Either you are going to kill the tree, or it is going to live, and do what someone suggested, and develop a callous that will end up damaging your fence. The only thing I can think of, which I remember from someone saving a tree that was vandalized, is to replace the bark in the cut area. You are going to have to do something like prune green branches from the top, and then cut them to wedge into the spot where you have removed part of the trunk. They will carry sap over the cut place, and protect the spot from the elements and fungi as much as possible. You probably are also going to want to use pruning tar, or the “bandages” (I’m not sure what the proper term is) that people wrap around grafted limbs until they heal.
I know someone who had a tree that was vandalized by someone cutting chunks out of it, and he saved it by wedging green twigs from the top into the cut parts. Eventually new bark grew in the places between the twigs, and callouses formed, but it still had indentations.
This sounds like a lot of work to me.
I’d either get rid of the tree, or do something else with the fence. If you really want to try cutting a chunk out of the tree, I’d consult a tree surgeon, rather then just taking a chain saw to it.
If it dies, and it dies sideways, so to speak, rotting where you cut it, but still being half alive, it will fall over, and may fall onto your fence, or your house. That’s the last thing you want.
Another thing you might want to think about is that if you are putting a fence that close to a tree, you are going to have to deal with the roots. You really might be better off getting someone professional to remove the tree, and the larger roots, so they won’t interfere with the fence.
When I had a house, the well had been dug 20 feet from a tree, which was probably a sapling when the well was dug, but it was a huge tree when I moved in (as a renter before I bought it), and the roots of the tree were starting to grow into the well house (one side was badly cracked). The landlords had to remove it before it got big enough either to destroy the well house, or interfere with the pump and the pipes. The people who had lived there before me were chronically late with the rent, so they had not reported things that were wrong with the house, and in the first six months I was there, I was constantly finding things overdue for repairs. The landlords actually ended up giving me a month’s free rent to vacate for two weeks (I could leave my stuff there) while they did some major repairs. They put in central air, among other things, and resunk the well. New pipes. Reshingled.
Anyway, you really don’t want to leave a tree that is already a problem, because they keep growing. If it’s in your way now, it is going to be MORE in your way next year.