No, you don't need a note from your doctor. I do think you need concurrence from your player, in the general sense, because you are in effect ending their season at this point, possibly messing up any performance bonuses, etc. If Kluber is not physically able to pitch at an MLB level ( much less AAA) then he would be likely to go along with any program that does not embarass him any further. In the hypothetical, a 60 day IL designation would seem better for him than a DFA tag. He'll get paid either way.
If a 60 day IL assignment for anyone did not require some degree of confirmation of physical ailment, then it would be too easy to solve roster issues by using them generously. I am sure it has been abused from time to time.