Mixing a RBR10 with ?

There's no other way to put this: you are sort of screwed.

Because of the way the "10" product was designed (a "Dual Band" rather than a "Tri Band" system), the only satellite compatible with the RBR10 router is the RBS10 satellite, which has no Ethernet port.

However, all is not lost.  Are you absolutely certain that the TV will not stream 4K video  over WiFi?  If the issue is signal strength, maybe an RBS10 in the same room as the TV would boost the signal enough.

Another solution might be to use either a WiFi to Ethernet adapter or a WiFi Extender with an Ethernet jack on it, such as the Netgear EX3700.