The Derek Zoolander Center for Kids Who Can't Read Good and Who Wanna Learn to Do Other Stuff Good Too. Hog’s Warts... no one will notice! Seriously... you could just use (Name) Nobles Academy or... Grand Academy of (Subject). These are kinda generic though.
I think court is also a justifiable word for scholarly proving grounds, considering Chinese styles that certainly resembles it.
Definition wise it's correct, but it doesn't really feel right for militaristic purposes to me at least. But you know the specifics, it's up to you.
Congress doesn't really work since it's more indicative of a formal meeting, and would really give the readers a false impression. It seems like you are trying to reinvent the wheel when there are ready made definitions that specifically address what you are trying to convey. Academy or institute seem like direct representations of the meaning you are trying to convey. Why not focus on the proper name and then see which might fit in better at the end. You could also just create a name for the school that is completely unique without any actual reference to school type in the proper name and simply insert the description into a sentence or two to convey to the reader the actual type of school/meaning An example: In the US there is a school known as "The Citadel" which has a full name of "The Military College of South Carolina" although no one ever uses the full name.
Ludus is the name of early schooling in Rome (albeit slightly younger than teenagers) and at same time Ludus Magnus was the name of the gladiator training facility. Then if you go further back to greece there is the Agoge (Spartian) and Gymnasium (Athens). Athens had 3 public schools, Academy, the Lyceum and the Cynosarges.
The reason why I thought Congress might fit is that it's not really schooling that they'll be doing. Not entirely. More than learning stuff, they get to form connections that might help them in their future. True they'll learn certain skills but that's secondary. Think an international exchange program where kids meet to learn together for a few weeks and learn about different cultures. It's sort of like that where formal education isn't really the aim. Though, Institute does sound well and seems fine for that.
Brainwashing facility. You think I’m joking, but this term isn’t per say wrong. To be taught something is to change the way a brain functions. Once you are taught math, you will automatically refer back to said math. Once you are taught how to read, you will automatically read when presented with words. The taught brain was forever changed through the education~
Classes in which history, civics, social studies are taught aren’t free of this either. In many cases, especially on the lower levels, their is a goal of installing a sense on patriotism. I won’t say that’s wrong, as long as it isn’t some sort of extreme fanatism, but that’s teaching someone how to think. That’s even more so a type of brain washing~