I’m not certain if there are mixers out there that can do what you’re describing that aren’t getting into pro broadcast prices, especially for 4 cameras.
You might find this thread helpful: How to convert full color ntsc video to lzx format?. You could run 4 cameras thru multiple genlocked scan converters as described and then you could freely patch your sources direct into your LZX system or your mixer. You don’t actually need Cadet III for this to work–RCA to 3.5mm adapters will do in a pinch.