How often do you do 75 percent water changes? 1/2 gallon is really small and ammonia will build up very quickly in that size container. You should be doing water changes every day with such a small tank, but really he needs a bigger container. It could be all of the ammonia that is making him sick.
What temperature is his tank? Is it heated? Bettas are tropical fish and need temperatures around 78-82; if their water is too cold they will become lethargic; this may also be the problem.