I like your idea. Let's find out something about what the Googlebot is "seeing" and log the POST requests in a database table.

Surely you can do this with GET with minor server trickery? (Has such been done already?)

Correct. The richness of the Javascript environment for Googlebot is probably a more significant discovery than the POST requests. I previously pretty much assumed that bots didn't really run Javacript. (Or at least that they would kill network access before running it)