I would enlarge the damage to try and make it rectangular so I can easily cut a patch to fit. That might be possible without deconstructing the joint, but might not. I would pick a piece of white oak that has as close to a grain-match as possible and darken it with garnet shellac to match before a final finish.
Ding Ding Ding! We have a winner!
I looked at the damage a little more closely and realized that, while the joint is currently holding, on the opposite side is where the glides are attached and the stress from pivoting and sliding the door in and out is concentrated. I am going to be taking this up to my daughter in NYC (happy birthday Laura) so if it eventually fails I won't be able to fix it for her. I decided to approach this sort of like a very thick veneer repair. I scribed a line with a marking gauge to mark a rectangular cutout and leveled the area by first using a fine Japanese saw and then a chisel.
I found a chunk of white oak that's been laying around for years and sliced off a 3/8 inch slab which I then resawed for about a 1/8" thick piece and cut a piece slightly wider than the cutout. I used a low angle block plane to dimension the width and also to add a bevel to match the joint. I left the thickness proud so I could sneak up on it once I had a tight fit.
I used my Kunz 100 to carefully level it using it more like a controlled chisel making skew cuts. I put masking tape around the piece so that I could stop before I cut into the surrounding wood. I did this before cutting it to length or gluing it in place so that the extra length could help stabilize the plane near the edge. Once the tape started to tear I knew to stop on that edge. The left side is higher so I continued working on that side until I hit the tape. I then removed the tape and use a chisel lying flat on the surrounding surface to nearly eliminate any raised edges. You can still feel it but I felt like this was good enough. I cut it to length and it was ready for glue up with hide glue. It was a nice tight fit and with the bevel sliding under the joint, it did not require much clamping.
Note that I used a gouge to better match the cove area not inside the joint. This was before I finished shaping the cove section and leveling.
I used some Mohawk finish repair markers to blend several different colors to get a fairly close match.
I am pretty happy with the result. Unless the light hits it just right or someone looks at it closely, I doubt that most people will be able to tell. I will probably apply a little shellac before remounting the door to the cabinet.
Thanks everyone for your input. It really helped. I considered all of your input, including doing nothing.
--Nathan, TX. Hire the lazy man. He may not do as much work but that's because he will find a better way.