One Windows Domain workstation can ping gateway but gets no internet access

You are right to not want to run multiple DHCP servers on the same network, that can end up giving very random results.

Right now it sounds like you have both wireless routers plugged into your main router by their internal ports, using them basically as access points instead of routers (just getting the wireless users connected into the main network and letting the main router do the real thinking).

If the wireless router the students are on allows you to setup your port blocking rules to specific addresses, you may be able to on that one set it to block all communication, then allow specific rules to your internal servers. Then connect that router to your main network by the Internet/WAN port on the router and enable DHCP on it. That will make it so that the students are segmented on their own network and can only access your main network routing through the wireless router, which has the rules setup on it.

So, you would have:
Internet Connection <-> Main Router WAN Port
Main Router Internal Port <-> Teacher Wireless Router Internal Port
Main Router Internal Port <-> Student Wireless Router WAN Port

Anyone connected to the main router or the teacher wireless will be in one network and have normal access. Anyone connected to the student wireless will be in another network behind the student wireless router and be subjected to the rules setup on that router.

Otherwise, you'd want to look into setting up a DMZ to separate the students from everyone else, then setup your specific rules they are allowed to do. Your current equipment may not support this, your main wired router would be the one to look at.

Since you are serving your DHCP from Windows and your clients are also Windows there might be better solution that you should be looking at.

When a system has multiple interfaces and multiple default gateways set, the method that can be used to force one to take preference is to set the metric.

Instead of messing around with DNS, set the options on your DHCP server to control the metric associated with the IP. Set the metric on you wireless LAN to have a larger value then the metric on the wired LAN. This does assume that your Windows DHCP server is serving both the wired and wireless networks and they have different scopes.

If this applies go into the scope options ant set up the Microsoft Default Route Metric Base.

If setting a metric via DHCP is not an option, then you can also set it directly on the TCP/IP properties for the wireless adapters on those computers. The interface with the lowest value is used by default.