I see, thanks for the clarity. if you connect one of your routers to the Switch and set it up as a router or Bridge, the others should be able to see it as well and connect via Wired Backhaul.
However, you would need to set the satellite devices up first before connecting to Ethernet.
Since you have a working network, you can always try and experiment.
This link may help you out
@WildBill seems to have been down the Bridge Mode path already.