Perhaps you would be better first looking at what they would expect you to prove, before considering the evidence to prove it. Perhaps "recognised" is the key here.
The first part is that the person needs to be registered disabled.
You mentioned Carer's allowance and among other things, that requires the claimant to be providing at least 35 hours a week of care - a big commitment. Even then the benefit is subject to stringent conditions which might prevent a claim, even if somebody were providing more than that level of care.https://www.gov.uk/carers-allowance
Presumably, if there were to be a close family relationship, and both people were living in the same household, this would not be an issue but it can't be so clear-cut if it's a friend who pops round. So, if somebody were to be registered disabled and living with their spouse, sibling, parent or adult child, that person would be a recognised carer, but I'd not bet on it.
I fancy that part of the problem is that the people at CyclingUK simply don't know and in any event cannot advise on insurance unless they are suitably qualified. Presumably, this policy is provided by an insurance company on behalf of CyclingUK. Perhaps that's the place to address your query as they would be the people investigating any claim.
You are certainly right to sort this out before it becomes an issue. (Sorry for not being more to the point.)
And FWIW, we did have something similar about the definition of "unwaged" in respect of the relevant membership concession. I don't remember anybody mentioning the potential insurance implications but if you think about it, they are pretty much the same.