Help me Recover this Redbud Tree (Photos Attached)

I've recently moved to Richland, WA (Eastern Washington) and have a struggling Redbud tree in the back yard. We are zone 7 (although, probably closer to a high 6). My soil is like beach sand when I dig. I have plenty of water available, but this is a desert (hot summers, cold winters with minimal precip ~7 inches a year). At the recommendation of the local nursery, I added soil amendments to try to help the tree (cutting out grass around the base) and mixing native soil with compost. I've also staked it to help with our high spring winds.

Attached are photo's of the tree. The first photo is unedited. The second photo includes labels of the branches for easier discussion. The third photo shows my proposed pruning cuts (in red). One thing of note is that branch A (right above the Y) didn’t produce any leaves this year and appears dead. The house was built in 2005 so I can assume that the tree was planted some time since then… I plan on pruning all of my trees in the late winter, but before spring (probably early February). Can you please give me advice? Thanks!