It does seem as if you have your ducks in a row about the doctors.
A while back I worked part time as a paralegal for an attorney who does Social Security Disability appeals. A couple of things to be aware of.
Your age matters. If you are under 50, there is a presumption you can be retrained and get work in a different field. If you are over 50, that assumption isn't there, so there's a greater chance of being approved.
They do NOT automatically deny everyone the first time around. People think that simply because so many of them don't have all of the paperwork and documentation needed. It takes a LOT of documentation. I know because I spent most of my job at one of the nearly Social Security offices photocopying extremely thick files.
Do read everything on the Social Security Administration website very carefully because it spells out very carefully everything you need to do, what documentation and proof you'll need.
If you do get denied, look for attorneys in your area who handle the appeals. You'll chat with them on the phone and they will figure out pretty quickly if an appeal will be successful. When I worked for my attorney, I was more than once in her office while she fielded such calls, and it was quite interesting. The ones she turned down, she always made a firm point of saying that hers was only one opinion, and they should consider talking to other attorneys. I will suggest that if you are turned down by two or three different ones, you may need to figure out what other kind of job you can possibly get.
A SSD appeals attorney gets paid some percentage of your award -- which be retroactive to the time you first applied I believe -- up to some specific dollar amount. You pay nothing out of pocket. And what the attorney gets is relatively finite, and will come out of the lump sum of the retroactive amount.
The worst thing for most people applying for disability is that they are not working and have no income. You are well ahead of things with the company insurance.
I hope you let us know how this turns out.