Does an imap proxy with full message list and message body caching exist?
I see two possible variations, one with on-demand caching (user clicks on a mailbox and a message list is fetched from "master" and cached for next click from another mail client / webmail connected to the proxy. Same for clicking on a message with message body cache). Another variation is auto-fetch, some kind of script or daemon that fetches messages on background.
for the first variation (fetch on demand) have a look at the dovecot imapc backend:
http://wiki2.dovecot.org/HowTo/ImapcProxy