He's following the process and I suspect the extension he's already received is to permit the process to go through to completion.
On the face of it, sounds like he has a good case--hope it works out for him.
On the positive side, if recalled the government has to pay for all the kids medical needs. On the down side you really have to "break the code" to get what you are entitled to on paper. The Army also has some "Screw the Soldier" policies regarding long term dependant hospital stays. The other services pay for the servicemember's living expenses if you have to take your child to a distant hospital. The Army pays up until the time you check your kid into the hospital then you go out of pocket.
This may have changed since last year, but I got stuck when I had to take a child for surgery in California because our "region" was covered by an Army office out of Fort Carson. My command put me on TAD funds instead. My only requirement was to put my research on the subject into a memo for our air wing staff.