Question
My exposure on my patio allows for all day sun, but will bougainvillia do well under a screen enclosure?

Answer
I am not sure where you live but Bougainvillea's are only perennials in Zone 9-11. So you must live in a warmer climate for them to survive the winter months.  They really do need Full sun to growth properly and flower.  So if your screen enclosure has full sun, which would mean it need to be south or west facing, it should fine.