Here is the answer to why they called themselves Hot Club de Paris, straight from the horses mouth.

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Our name of course comes from Django, and is a Django reference.
However, as far as I'm aware there was never a band in existence
called "Hot Club de Paris"... It's merely a play on The Quintette du
Hot Club de France... so yeh, why not? We all like Jazz, and we play
quite progressive music, in that it involves a lot of time signatures
and non standard arrangements... So why not make reference to
something that was pretty progressive in the first place?!

Matthew - member of Hot Club de Paris <<