Linewize would be the go!

hashbrown: ... Deploy a dns filtering product that lets you block games as a category.

dfnt: ... you could look at OpenDNS, pretty sure you can block by categories/domains etc, then setup NAT to force all DNS to the router

firefuze, hashbrown & dfnt, say hello to each other. The Vigor modems will let you specify your own DNS server, so easy enough to use OpenDNS et. el. 

Going against hashbrowns advice, if you have a Raspberry Pi handy, install Pi-Hole. Not only will you get the benefit of blocking ads, but blocking specific domains is as easy as clicking the 'Blacklist' button. Image copied from Scott Helme.

Edit: spelling

The best way is with web content filtering , at the router/firewall
https://www.draytek.co.uk/products/accessories/web-content-filtering

but
- it isnt free (annual cost)
- you'll need a more expensive router/firewall

Or, put some speed/bandwith limit on the students access, making it too slow/laggy for gaming

firefuze:

 

Dolts:

 

Do the students have a separate SSID? Limit the bandwidth to an unplayable rate.

 

 

 

 

Students do have a separate SSID which is currently rate limited, however if I reduce it further it would negatively impact those students whom are using the internet for genuine reasons

 

Does your Wifi have per IP bandwidth limiting. It's quite common to be able to do this on modern wifi (and firewalls), ie assign 10Mbps to "students" SSID with a per IP limit of 512Kbps (or whatever you want). Then each user is limited to a maximum and cannot impact other units.