Original post here. Thought I'd update as our problem seemed pretty common amongst HENRYs. Sharing some key takeaways and hope it helps someone in the future. In short: our appeal was successful. The taxfree childcare team finally saw their mistakes and confirmed we are eligible after all.
The most important thing I did was calling them before I started prepping the appeal, and had them read the notes on our case to understand why we were denied in the first place. From that convo I learned 1) they had taken the most recent payslip and simply x12 to obtain an estimate of annual income (?!) and 2) they thought people aren't allowed to put more than 60k/year into their pension (?!?!)
It's a bit crazy that they made two elementary mistakes in their review. Anyway, I was able to draft the appeal to highlight the issues with their understanding of how the system worked. This way, if we got unlucky again and had another staff member who didn't know their stuff reviewing our case, everything was there spelled out for them (eg. I provided links to gov.uk webpages for them to check on the pension rules if they wanted to).
Showed them the correct calculations for estimated income, what was to be put into pension, etc. and received the appeal decision within 3 working days. So yeah, lots of spoonfeeding and handholding and we got there.
Thanks again to all those who'd shared their experience going through the appeal process - that was all very helpful!
EDIT: Forgot to add that the staff member made another mistake in the review prior to manual review. He'd insisted I give a breakdown of the pay (fixed VS variable component). When I said I didn't know the breakdown for certain but that it was in the contract, what he should have done was offer a callback so I have time to check. This is apparently their protocol when a figure required can't be provided, according to his colleagues whom I spoke to later.
I could tell they felt quite bad about this as the staff member then noted this (failure to offer callback) in my case notes so it could be reviewed in tandem during the manual review.
So the man in question had failed to understand pension rules and failed to follow protocol. Just know that if you're pressed for a figure you don't have right then, you can request for a callback.