The cheapest way to do hdmi to RCA is one of the generic hdmi to VGA adapters on ebay or one of the chinese marketplaces (like this one.HDMI goes in, and you've got a 3.5mm audio output.

In reviewing many product pages, I found a single ~$70 HDMI switch w/ RCA audio out that *claimed* to do what I wanted but was from a no-name brand on a no-name website; this was sketchy enough ...

For example, your TV likely has HDMI 1.4 connections, which max out at 3,820x2,160-pixel resolution at 30 frames per second. If you've gotten a new 4K monitor, you're limited to 30fps.