Are there any "Clubs" in Liverpool City Centre that have a similar format to that of Piccadilly & Cosmopolitan in Manchester?
No.
The only Non-Asian Parlour is Temple Spa 21
External Link/Members OnlyListed on Google as Spa 21.
Services include tantric, naturist and nuru massage.
Compared to the nearby Chinese/Thai places, it's expensive.
However it's nothing like the Manchester venues you mentioned.
It's years since I've been and well before joining this site but the fact I never returned speaks volumes.
The reviews aren't very encouraging.