Ok - of course 20 seconds after posting this, I find an answer on superuser. Probably where I should have posted to begin with (I swear I googled before posting but apparently not enough).
SU post: https://superuser.com/questions/43238/activesync-owa-desktop-client
Answer: http://davmail.sourceforge.net/
This appears to be a POP3/IMAP and SMTP bridge to activesync which is exactly what I want. If it works as claimed, I can point any mail client to it and it will sync my mail in two directions so I can send messages that will pass through the system and see all the messages that come in.
Update 2013: I've been using this DavMail OSS software for a few years now and it's really great. For retrieving mail from relatively locked-down Exchange servers, that only permit OWA access, it very easily allows me to "break out" of the box the IT guys want to put me in, and use rational tools and protocols like IMAP, POP, SMTP and Thunderbird to send/retrieve mail from the Exchange server. If I were really motivated, I could get this running on a public server, and serve my mail so that Gmail could poll it, and bridge Exchange and Gmail, but of course I'd have to pay more attention to security.
I haven't had a ton of luck getting the Calendar portions of the system working, primarily because I can't find a caldav client (on Windows) that I like - I think the DavMail part works as well as can be expected.