Which tables to create according to relations
It isn't necessary to have a surrogate primary key on your clubs_chains table.

The combination of the two foreign keys in clubs_chains is adequate for the primary key.

You can use a foreign key constraint to ensure that your clubs_chains_paymethod table references an existing record in clubs_chains using the compound primary key.  This might be helpful since it would allow more direct joins between clubs_chains_paymethod and your clubs and chains tables, without sacrificing any referential integrity.

On the other hand, some people just love using a surrogate primary key, even on intersection tables.  If that is your data modelling style, then it's OK too.