Proxy-Authenticate / Proxy-Authorization are defined as hop-by-hop headers in RFC2617
"Both the Proxy-Authenticate and the Proxy-Authorization header fields are hop-by-hop headers (see section 13.5.1 of )."
Hop-by-hop headers are defined as:
"Hop-by-hop headers, which are meaningful only for a single transport-level connection, and are not stored by caches or forwarded by proxies."
Although you may be able to leverage Cookie Auth on the second proxy, because it uses WWW-Authenticate, it may be overly complicated.
If they are MWG's at both locations, you could have the first proxy insert a username header and the second proxy read that username and use it: