Damn, thanks for that super-detailed breakdown.
I was (naively) hoping that using an MFi Lightning-to-Type-C cable would solve the problem, but I guess the BTR3 itself needs to be MFi certified. Do you know of any Lightning-to-Type-C cables with hacked chips that might make things work similarly to the CCK?
Actually the easiest possible solution I can think of might be to take an official CCK and mod it with a Type-C connector. There are 'naked' PCB only Lightning connector on Taobao that claims to have the official MFi chip inside though I am not sure how real that is. Those can be turned into a Lightning-to-Type-C cable fairly easily.
There are a few seller on Taobao that sold the latter, so it probably will work. Not really sure Aliexpress has those one or not.
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