The Linksys EA600 needs to be put into bridge mode. If the router has that specific setting, it should work. If that setting isn't shown, you should turn of DHCP. The EA600 is going to try and assign IP addresses, which will conflict with the other router. There are other reasons why your existing hardware isn't optimal.
Two much better options are: a) not hardwiring the apple tv or b) purchasing Google Mesh or a similar mesh router setup. These new mesh setups solve for the absurd technical hurdles that have plagued home routers forever. You can hardwire the mesh pucks or use them wirelessly and the configuration is managed via phone.
