There is a general rule in English, that "proper names" do not require an article (such as "the" or "a/an"), but normal nouns do. So, to change your fantasy institutions to real world institutions, we would say:
He was studying
inat Ohio State University with the hopes of transferring to Yale.
We don't use the in this sentence, because the names of these universities are considered their "proper names". What would it look like if we didn't use the proper names?
He was studying at the state university in Ohio with the hopes of transferring to an Ivy League school.
Here, state is an adjective modifying the noun university, and is not a proper name, so an article is required.
Where it gets tricky is when the proper name could also be seen as an adjective/noun pair.
He was studying at [the?] Springfield Community College...
Here, it could be ambiguous whether or not to use "the". You could read Springfield Community College as meaning "the Community College that exists in the town of Springfield", in which case it's an adjective/noun pair and you must use an article. Or, you could read it as a proper name, just Springfield Community College, a proper name, where no article should be used. And in fact, as the writer, you could successfully choose either option, and be correct either way.
So as the author of your story, you can choose either form, and be correct. However, each form has a slightly different meaning. Without the "the", you are saying that this is the actual name of the magical institution. With the "the", you are saying that this is a description of the institution, but not its actual name.