In Orlando, this is what I do to my collection of around 20 or 30 (or so lol) bougies:
Method: Cut the tree back. Apply Bougain. Go. Bracts form. When bracts die, or you're done, chop it back again wire out what you want to.
Bougain does not burn a bougainvillea, it's almost impossible to. If you are using some 20-20-20 or even another odd feed, it doesnt provide the bougainvillea the nutrients that encourage more flowering. Instead, those fertilizers encourage more growth. You'll end up just with thick branches in weird places. Bougain. Swear by it.
Of course, the standard disclaimers, I'm in Florida where they naturally grow, you are in Illinois. We use coarse soil here, well draining, and keep our bougies trunks dry but the roots moist, never wet feet never bone dry. This method repeatedly encourages four or five wonderful blooms each year, as well as develops the tree.
Additionally, Bougainvilleas require intense sunlight. I dont think Irish people tan in Illinois. Here, they are all white because have have a half inch of sun screen coating them like toothpaste. Bougies require full sun to fully bloom. If you Want to test, turn tree sideways so only half gets full sun, half gets part sun. You see color one side, green other. Sun sun sun sun sun.
There are many varieties of bougies too, so sometimes my advice may change for varieties such as the Pink Pixie, or a Royal Grape, for example, each work the same but slightly different.
The "WIN THE EPCOT ONE DAY BONSAI SHOW" award trick is simple. Chop back, Bougain 6 weeks before show. Tree wins people choice. Golden Mickey Mouse Statue. All bracts/flowers.