The part of the establishment which had the highest snob count in the early/mid-60s (and even today) was probably the art world, and Lennon wasn't exactly backward in coming forward there.
Ditto The Stones.
See the Guinness family, Tara Browne ('A Day In The Life'), Miles, Jane Asher, Robert Fraser etc. etc.
I'm sure Lennon's initial take on Brian Epstein and George Martin could have been confrontational and, while not exactly snobs, they were definitely posh (more posh than Ringo!), and almost certainly more posh than 90% of that Royal Command audience.
My mum appeared in the Royal Command many, many, many years before The Beatles and her recollection is that most of the audience where family members of the performers (two tickets per performer), and it would have been the same in 1963, and they all would have 'dressed up' for the night.
I don't think John was being deliberately spiteful because, mercifully, in 1963, the casual antagonism we take for granted today, and not too many years after 1963, was not so prevalent or acceptable.