Perhaps making it private, so only members of your club can participate? I'm unsure if such an ability exists. The games would still be posted, even if the jam isn't public. I've seen plenty of games that state "made for college project."
Assets are a good way to limit things, so you know they were made for your game jam. Most game jams go with a theme, or 3 themes "dark," "pink," "world shifting." Bonus points for including all 3!
As for the 1 week, seems fine. I've seen others argue 1 month is lower commitment, if you're busy with stuff for 3 weeks, you might not be busy that one week. Then again, people like me, might work that entire month.
Best of luck to you and your game development club.