If they keep it as Fumizuki, I think that it would be okay to use it without "Academy" added to it. Though having Fumizuki appear during its first mention as "Fumizuki Academy" might be better if it is possible. Also, if I am correct, proper nouns normally stay mostly the same even when translated. If they did go with "July Academy", they could probably just abbreviate it. With the example of "Computer School", since "computer" is recognized as a common noun that would be somewhat odd to see by itself, but it could still work since it is "Computer" rather than "computer", which shows that it is being used as a proper noun rather than as a common noun. Also, people often use the names of organizations without anything else (e.g. Harvard, Yale, and Princeton; Sony, Microsoft, and Nintendo) and still know what they are. With "July Academy" however, it is more complicated, as July is already a proper noun that commonly refers to a month and so could possibly cause confusion. Though, as I said, they could just go with a shorter form of it.