Instead of hacking the firewall, use social engineering.
You would be surprised how often this works. IT guards hate distractions but love educational tools.
Alternative solution
User education
Monitoring
Balatro is a popular roguelike deck-building game (rated PEGI 12 / ESRB Teen). Recently, it has appeared on “unblocked games” proxy sites. This report examines what “unblocked Balatro” refers to, the risks of accessing it via third‑party proxy sites, and recommendations for school/network administrators.