You should understand that fat itself doesn't cause bile to flow. It is the hormone called cholecystokinin (CCK). When food containing fat (and amino acids) enters the digestive tract, it stimulates the secretion of cholecystokinin (CCK) from cells of the duodenum and jejunum. It is accepted that many sulphurous foods also cause the gallbladder to contract, such as cabbage for example. Other hormones inhibit release of CCK. So when you eat a complex meal of fat, protein, and carbohydrates there cause hormones to be produced and the proportion of these determine how much bile is delivered and how quickly, to the digestive tract from the liver and gallbladder, and how these are mixed with juices from the pancreas.
The gallbladder will be fully charged in a 24 hour cycle so there is no need to do any more. Each evening the gallbladder will contract slightly and then expand each morning as part of the normal digestive process.