to be more accurate about the pool club in Paris , here is the (poor) list of the carom clubs of Paris (downtown):
CLICHY MONTMARTRE BILLARD CLUB 84 RUE DE CLICHY
01 48 78 32 85
clichy-montmartre@wanadoo.fr PARIS
BILLARD CLUB PARISIEN 1 RUE PIERRE LESCOT 01 40 13 95 92
nadine.treille@orange.fr PARIS
SHOOT AGAIN PARIS NATION 9, CITÉ DEBERGUE 01 43 43 32 10
shootagainparis@gmail.com PARIS
my advice if you do prefer to play only in downtown/in Paris : visit the Clichy academie, and play at the Nation billard club

. not easy to play at the place clichy academie, not really "player friendly"
around Paris ( easy access with Metro / train ) :
Academie DE BILLARD COURBEVOIE-LA DEFENSE 4 ALLÉE DE CHAMPAGNE 01 47 88 83 75
billard-courbevoie@wanadoo.fr COURBEVOIE
ASS. BILLARD AMATEUR DE SAINT MAUR 34 AV. DE LA RÉPUBLIQUE 01 48 83 59 44
abasm@free.fr ST MAUR DES FOSSES
Billard Club du Kremlin (BCK) 23,25 AVENUE DE FONTAINEBLEAU 01 46 72 51 51
billardclub.kremlin@gmail.com LE KREMLIN BICETRE
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seriously , if you really wanna play carom and to meet players during a rip to Paris, i suggest you to visit :
- the Courbevoie academie . You'll find a ton of "small games" players (straight rail/balkline) and do not speak about 3C , because the house pro there is Jerome Barbeillon , >1.2 annual average .
- the Kremlin, bicetre club which is a superb poolroom, with a crowd of pool tables too . player friendly .
IMHO those are the best poolrooms when it comes to the very near Paris , but it's just my modest personnal opinion.