Could you have a server that knows where the dynamic url is at all times, and provides a redirect? So I'd hit up lwsh.me and it would redirect me to <https://plus.google.com/hangouts/_/etc> ...that would create an effectively permanent url, even though the hangout itself would change urls.

Looking at the Hangouts API, it appears that when the app is initialized you could call getHangoutUrl() and then pipe it back to the server. This could probably be used in a pretty dynamic manner too... like whenever anyone uses the app, it connects with the main server and adds that chat to the list of active chats...