I use Construct 3 in a college classroom with older students (17 to 22) and have found that the cloud storage system makes it relatively easy to keep an eye on student progress. This is how I do it:
a) I subscribe to a paid Dropbox account that offers 1TB storage.
b) Dropbox is installed locally on all of the lab computers, with all computers pointing to the same Dropbox account. I have created project folders for each student inside that Dropbox account. (Note: These folders are not stored inside the Apps folder that C3 uses for cloud storage. These folders are used for all of the students' other digital production work.)
c) Within the Dropbox Apps folder that C3 uses for cloud storage, I have created a second set of student project folders. I ask students to make sure that *all* of their C3 projects are saved and clearly labeled in these folders. If they encounter problems along the way, I can easily access their folders and files through Dropbox from anywhere in the world.
If it is a hassle to install a local version of Dropbox on all of the lab computers at your school, you could just skip step b.
As for registering students in the class, this is part of the educational package that C3 offers. At the beginning of the semester, I assigned a license (or seat) to each student. It's easy to do, and -- in my opinion -- reasonably priced. If it was much more expensive, I'm not sure I could convince my school to cover the costs, but C3 is currently an excellent value proposition in the classroom.