What’s the probability of remaining at a duty station/assignment after your year of a compassionate reassignment?

Look into the medical stabilization.
Quickly: I put in for a compassionate reassignment to where I currently am at and they happened to have a slot for my MOS and it's a 3 year assignment. My 3 years was supposed to be up in January but, knowing last year around April, May my daughter still hadn't gotten a new kidney, I looked around for other options to see if I could extend. The Army SEL at the time told me about a MILPER he saw about medical stabilization. Now what it doesn't say in the MILPER is that you can get up to four years authorized to stay at your assignment - but it's 4 years from the start of EFMP. So I got that approved but my daughter's EFMP started in Aug 2014...so I got extended eight months where I am and really 10 because I PCS in October now.

MILPER Message Number 17-093 Stabilization of Soldiers and Military Families with Special Needs
4. Criteria. One or more of the below must be met in order to establish eligibility for stabilization under this program:

a. A permanent change of station (PCS) move would disrupt access to necessary medical or educational services that would place the EFM’s health, safety, or development in jeopardy.
b. The EFM has a diagnosis that requires extensive support from TRICARE, state, and local resources that would be difficult to replicate and/or reacquire in a timely manner.
c. The Family has multiple EFM’s receiving a combination of TRICARE, state, and local services that would be difficult to replicate and/or reacquire in a timely manner.

I don't know if that could help you at all. Otherwise branch would be your only other option.

Find a unit on Bragg that you know has your MOS. Get on the CSM's calendar and request a Paragraph, Position, and line number.

I'd try and find my own slot before hitting up branch. Yes, branch has to fill slots, but, they also like to move people around. Find a slot for yourself, get the connections in place and THEN tell branch you have a solution to a problem they didn't even know they had. If someone comes to me with a solution to a problem before I know about the problem, 9/10 I'll take that solution.