Ethernet and Airport Problems
Solution 1:
I'm guessing here, but it your network settings may have the ethernet interface prioritized above the AirPort (this is the default setting), meaning that when both are connected it'll try to reach the internet through the ethernet connection. You can check this easily be looking at the order the network "services" are listed in the sidebar of the Network system preference pane. If Ethernet is prioritized above AirPort, change it by selecting Set Service Order from the action (gear) pop-up menu below the service list, then dragging AirPort above Ethernet. Note that you need to click the Apply button for the change to take effect.
Solution 2:
I suspect the issue might be that your routers probably have the same IP address on both networks. So if your printer router and your internet router both have an IP address of 192.168.1.1 then your Mac may not know which interface to choose for traffic to that network. So it will probably pick the network which is listed first on the list in System Preferences > Network.
You could try reconfiguring the printer router to have an IP address of 192.168.2.1 and setting it to use DHCP with a subnet mask of 255.255.255.0. Then your Mac could have an IP address of 192.168.2.19 and it would know that traffic for the printer should go out on the wired connection.
Just as a note, you can change the order your Mac selects networks by clicking the gear wheel button beneath the interface list and choosing 'Set Service Order' but I don't think that's going to help you in this case.
Solution 3:
The ideal solution is indeed to simplify your home network by only having a single router (likely the existing Airport router). Using a single router and it's DHCP server will assure that all of your home devices are assigned IP addresses that are within the same IP subnet and can therefore all communicate with one another.
You could consider replacing the second router near the Canon copier with a WiFi-to-Ethernet bridge to provide available wired Ethernet ports but simply bridge them to the existing Airport network. A bridge keeps all the devices on the same IP subnet (unlike a router in it's default configuration). If the router near the copier also has wireless capabilities you may be able to configure it as a bridge instead of a router by looking through it's web configuration screen.
Once all of your wired and wireless devices are on the same IP subnet and using the same router (default gateway) then they will all be able to both see each other on the network and also get to the Internet.