No he does not need to see the letter as proof of your circumstances. So long as you have children under the age of 6, which you do, and that he does not have business reasons as set out by the flexible working regulations to refuse your request (which the fact of seeing the letter would not change so I would assume he does not) then there is no legal right for him to make this request.
It is up to you whether you raise this with him or not in the form of a grievance. If you can get hold of the letter reasonably easily then it might not be worth bringing it up as he may then find a valid business reason to refuse.
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