How can I get Tasker to set my Google Chat Status?
I found a plugin! When I first looked, I didn't realize Locale plugins were compatible with Tasker. The plugin is called "Locale GTalk Plug-in" by Billy Francisco. I have a Galaxy Nexus with ICS, and it appears to be working fine; my status was set properly by a Task.

There is a nearly identically named app, "Locale GTalk Plug-in Free" by Willem Stoker; reviews warned it includes Air Push advertising, so I did not install or test it.