The area you're thinking of between the BC and BCVs is called the Solarium. Generally speaking, it can be quiet but there can be stream of foot traffic going through there and also is used for overflow seating during breakfast/lunch.
I'd recommend heading over to the villas to the Drawing Room. It's just off the lobby to the right as you enter the villas.
It does have a pool table and TV in there, but I have rarely seen anyone in there mid-morning or early afternoon. You also may want to check out one of the Leisure [Quiet] Pools for a table with an umbrella.
Is this Pool Table available to anyone staying at YC and/or BC? Is there any cost?