HDMI Switcher for analogue audio?
No, it isn't possible.

An HDMI-to-RCA "cable" is really an active circuit that decodes the audio from the digital HDMI stream and converts it to analog. It takes advantage of the fact that the HDMI connector also has power on it.

The switch itself is active circuitry that is designed specifically to regenerate the high-speed serial data streams used in HDMI. There is no analog path through the switch.

On the input side, you would need a corresponding "RCA-to-HDMI" converter that digitizes the analog audio and encodes it into an HDMI stream. I don't think anyone makes these commercially, although there might be units that accept analog video + audio, and you could just ignore the video part.

But obviously, this is getting far away from your "lowest price" goal. You'd be far better served by getting something that's designed to switch analog audio directly. I'm not going to make any specific recommendations, because that would be off-topic here.