I would say that 'very' is an adverb modifying the verb 'dare'. It reminds me of the Hellenistic Greek construction when 'apo' is added to a verb and extends the meaning or turns the meaning into another dimension.
Apo is like a funnel. The stem is the original verb and apo then makes it expand outwards massively, inflating the concept.
Adding 'very' to a verb (or noun, see later) is a clever device, in my view.
The supposed joke, I understand, was that the original comedian never publicised his preference officially, though it was no secret privately. The second comedian used the original catchprase in contexts where the speaker's preference was being assumed, and he was expressing outrage at the suggestion. "How very dare you ! Come on, Edwin - we're leaving !"
The female comedian dressed up as a man to carry out the joke, underlining the fact that it was aimed, particularly, at men.
It was a comment on people who live one way privately and present something else to the public.
The comedian portrayed other, similar characters, one being the lady who constantly broke wind in a car and pretended she had not; and one being the grandmother who would express gratitude for presents but, once the donor had left, would enrage at the 'liberty' of being given something far less than she wanted.
In those cases, she dressed as a woman.
Famously, she portrayed a teenage schoolgirl who cared not a jot for what anyone thought of her and, in this case, the catchphrase 'Am I bovvered' (bothered) was publicised by Tony Blair, then Prime Minister of the UK, in an Xmas special on TV.
The two catchphrases 'How very dare you !' and 'Am I bovvered' are a comment on either society in its generations generally, or current society in particular - hypocrisy one the one hand and blatant rebellion on the other.
Perhaps the comedian is trying to tell us to be what we are, wherever we are, neither to pose nor to protest.
Bole - to throw; specifically, to cast a small net over a single large fish (see the South American bolas).
Apobole - to cast away. That is, to throw - but never to retrieve.
The Urban Dictionary carries a quote :
'How very Glenn Beck of you',
Glen Beck being famous for publicising an ailment as a reason for some of his behaviour but the publicised claim being described as 'pseudoscience'. Again, the implication 'very Glenn Beck' being that someone is one thing privately and another in public.
Glen Beck - Wikipedia
In this case there is an inversion, for it was Glenn Beck who expressed himself against criticism (how very !) and was revealed to be posturing. So the 'very' is used as an adjective to describe Glenn Beck. And the whole description 'very Glenn Beck' is then applied to someone else as a comment on their behaviour.
An interesting exercise in how language has developed - an original catchphrase, borrowed and turned back on its originator, then transferred to describe someone else, and inverted to apply to yet another party.
Thus the word 'very' has taken on a kind of life of its own.
All we have to do now is to use it as an adjective, no verb required, and to say 'how very of you'; and we will be understood.