How do you like the phrase ‘an utterly moot exercise’ to describe a completely fruitless endeavour? I just came up with the phrase and was surprised to find one hit when I googled it.
It seems consistent with the BrE definition of moot at wiktionary, but I’d defer to a British ear on that.
As to the the AmE definition of “debatable but of no practical relevance,” you know from the beginning that the theoretical exercise won’t bear fruit. The intensifier “utterly” would seem to have less applicability as I’m not sure there are degrees of mootness. But, what the heck.