I finally got an estimate back from a fencing contractor. My next step was to go around talking to the surrounding neighbors to show them a picture of the projected new fence and ask for their input. The neighbors on one side who don't own a dog seem to think they prefer the wooden fence for privacy, so I may have to leave that one in place. The other two neighbors are still thinking about it.
For reference, I took this Google Earth screen capture of my property. It's kind of amazing how small my garden area looks, compared to how much I know is crammed in there.
I spent several hours on Monday trimming branches from the tree along the back fence line, then chopping them up and laying them down as mulch around the sakura tree's drip line. I don't know if that will help it recover, but at least it can't hurt. If I get the go-ahead from the neighbors to proceed with the new fence, I'll have to make sure there aren't any branches obstructing the fence area anyway.