As you note, in "Theirs not to reason why", the word 'reason' is actually used as a verb.

I guess the simple word 'ask' didn't have the right number of syllables. It was used 'solely for prosodic reasons' - as Tom Lehrer would have said.

And there is nothing grammatically wrong with "May I ask why?"

So, while it may still be a sin to say "the reason why", Alf the Lord T is not guilty.