As you may recall, "Let It Be" was being documentarized as a contractual obligation, despite none of the Beatles wanting to make a film. They were being pressured to come up with film and song ideas at a low point in their creativity.
My guess, based on that context, is that John blew off steam during the sessions from time to time by jokingly throwing around nonsensical song and film titles. He did something similar when ending a jam session by saying, in a mocking schoolteacher voice, "That was Can You Dig It by Georgie Wood!"
"I Dig a Pygmy" is of course a riff of the name "I Dig a Pony", a song that he was working on at the time. In the very much less politically correct times of 1960s Britain, "Pygmies" were a cartoonish stereotype of African bushmen, which you might have seen in old Bugs Bunny cartoons.
Charles Hawtry was a 60s comedic actor who starred in lots of cheesy movies. "I Dig a Pigmy" would have been just the sort of cheesy pun-name for the type of movie that he would have appeared in.
"Deaf Aids" is/was a British way of saying hearing aids. The Beatles mockingly called their studio earphones "deaf aids".
"Charles Hawtrey and the Deaf Aids" was John making up a jokey name for a band that would play a cheesy novelty song like "I Dig a Pigmy". The name was a parody of a by-then-ridiculously-old-fashioned band-naming convention that they grew up with, ie.,"[Someone] and the [Somethings]".
"Phase One, in Which..." is a reference to an old style of naming chapters for books that John would have grown up reading. For example, in "Around the World in 80 Days", chapter two is called "Chapter Two In Which Passepartout Is Convinced That He Has Found His Ideal".
"Doris Gets Her Oats" -- John improvised the name "Doris" as a character for his made-up old-time story / film (ie., like Wizard of Oz Doris). "Gets her oats" is a mock old-fashioned euphemism to describe what we might today call "banging". This was humoristically added as a saucy twist to John's made-up cheesy film/story/song.
So in sum, "I Dig a Pigmy by Charles Hawtrey and the Deaf Aids, Phase One in which Doris Gets Her Oats!" is John jokingly making up a fake song / film title for the Rock Doc that they were being forced into making, using a bunch of cheesy, out-of-date band/film/book naming conventions and pop culture references to make it sound as ridiculous as possible, for comedic effect.