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a HDMI to AV converter, including the usb power supply and the RCA to SCART cable, should be sufficient, as long there is already a wireless network available. I have done this with a firestick, but be aware the image is not really good, especially if you are using some cheap converter typically available for about ~15 €.
I do not get what you want to show us with the router on the left side and especially the "wired" (?) connection of the chromecast to the "USB" (?) connection on the left. The video/audio information is usually streamed via wifi if you do not a dedicated LAN adaper, which I can not spot in your sketch.