Thanks a lot for your response.
From what you are saying I understand another option:

current server section - Server A
copy the same configuration and use some other spare drive as a proxy_cache_path configure that as - Server B

Configure both server sections to listen on unix socket instead of network.

Create a third server C configuration that will listen on network and will proxy_path to Server A with proxy_intercept_errors on and error_page served from location that will proxy_path to Server B.

Is something like this supposed to work? Or it would be better to have the there completely independent configuration (like to use some other software for server C).

(I'm asking because I did something like that i the past and it broke a bad way. It started serving 500 to everyone. I kind of fear to try it in production again.)