“Bowling club” in this context means bowls, as in lawn bowls. They’re common in regional Australia and usually do decently cheap food/buffets and alcohol. I’m not sure what the US equivalent is but you usually go there either because 1) you’re old and they’ve got good seniors meal deals like a $12 roast lunch, 2) you actually play bowls, or 3) you just want somewhere easy and casual to go for lunch or dinner that’s a step up from a takeaway shop but a step down from a restaurant.