Mr Commissioner Jacobs held in CH4424/2004 that a determination of incapacity was a matter for the secretary of state . His decision was in the context of rent restrictions under old Regulation 11 but I dont see that the principle should be any different in the context of childcare costs.
The simplest was for this claimant to get the secretary of state to make the necessary determination would be for him to claim incapacity benefit. He will be entitled to IB if he fullfills the incapacity criteria under the own occupation test or the personal capability asessment as appropriate, and he has paid sufficient national insurance contributions in the relevant tax years.
If he is not entitled to IB because his NI contributions are not up to date, he could still be entitled to national insurance credits, so making the claim would still benefit him notwithstanding that he needs the incapacity determination in order for you to award the childcare costs.