Double or single plural?

Solution 1:

In this context, club is an attributive noun, which should appear in its base case. You lose the information about whether you have one or many clubs. This can be inferred from the context or reinserted explicitly.

Club modifies booking. You have a certain kind of booking; each one is called a club booking. Since you have multiple of these, you use the plural, bookings.

Putting this together, you’d call them club bookings:

• Club bookings close tonight at 10pm.

If you want to note that you have more than one club, you can use the phrase “bookings for all clubs”, which preserves the plurality of both clubs and bookings.