@sloopercat - I'm not sure if you are helping your daughter, your mother, another relative, or a friend, but if she worked as a teacher for 27 years until she couldn't anymore due to worsening ME/CFS, she certainly deserves to have her long-term disability application approved. I'm sorry she is in this situation.
Unfortunately, I have not had to go through an appeal myself, so I cannot offer you an example of a successful appeal.
One big advantage of hiring a lawyer for the appeal is the signal it sends to the disability insurer that if it upholds its denial on appeal, you have lawyered up and could file a lawsuit. Though ERISA offers protections to private disability insurers (e.g. no punitive damages, a narrow set of criteria for a federal judge to reverse the denial), those insurers don't want to spend money paying their lawyers to prep for and argue a court case that they are likely to lose. Disability consultant Linda Nee, who writes a regular blog about private disability insurance, strongly recommends hiring a lawyer for the appeals process for this reason, as well as to avoid mistakes.
If you do most of the legwork in terms of collecting letters/reports/records, that will cut down on hourly legal fees. The lawyer's role, then, is just to suggest edits to documents that will strenghten the appeal, and to submit the documents on your behalf.
Off the top of my head, two other suggestions come to mind:
I'm not sure who conducted the 2-day CPET, but if it wasn't Betsy Keller at Ithaca College, you could send her an email (just Google her to find her email address) and ask if she can write a letter that explains in plain language why the CPET results prove both objective physical illness (not deconditioning or a psychiatric problem) and disability. Having a second renowned expert weigh in would increase the strength of the appeal.
You also might try to get in touch with Dr. Richard Podell. He is a retired ME/CFS physician, but as of a couple of years ago, he offered to serve as a consultant, writing letters for disability applicants with ME/CFS after conducting phone interviews and reviewing medical records and test reports. Gudrun Lange knows him, I think, so she might be able to provide you with his contact information if he's still working as a consultant. Having a second physician submit a strongly worded letter would strengthen the appeal.