They definitely appreciate being well-fed at this time of year. And at least five hours of full sun/light a day.

With your soil - is it actually dry below the top inch of soil? Just wondering in case there is some sort of pan below the surface that could be preventing the good drainage Bougainvillea enjoys so much. If it's present, you might need to take out the plant and dig down to improve the drainage - or build a mound so the roots can spread out without drowning.

Feeding: after you've watered you could use a balanced liquid feed that also provides micro-nutrients as well as the usual nitrogen-potassium-phosphorus once a month. And mulch with good compost (bought, if you don't already make your own). If you eat bananas at your house then chopped skins (take off those awful little plastic labels first, please) can be added to the mulch.

My guess, from the picture, is that it's hungry and needing both potash and phosphorus. Liquid feed on damp soil plus a good mulch will help in this regard. I'd avoid foliar feeding, though, in case it encourages moulds.

I would definitely say you are watering too much. You are only supposed to water when the leaves look wilted. Also, I bet it's not getting enough sun. Bougainvilleas thrive on neglect.