I'd suggest perhaps one of those toddler gate things you can buy, literally expands to fill the doorway, looks like a little pool fence. don't know how effective it would be on a 3 year old though. or where you would actually purchase one.
other than that, ikea sell some reasonable display cabinets with locks, though to have enough to display what sounds like an impressive collection, you might have to spend several hundred, close to 1000 I'm guessing.
The other option is a motion sensetive sentry gun with a 10000 paintball hopper.![]()