It turns out that the problems with the AutocompleteExtender actually go even deeper than the dispose problem. Or not as deep, depending on how you think about it. At some point in my effort to make my website pretty, the problem of the extender not extending after a partial postback reappeared. After spending a large, large amount of time on this, I found reference to another bug in the AJAXControlToolKit that happens if you programmatically set the focus back to the textbox associated with the extender control after doing a postback (partial or not).
I removed the set focus statement, and I am back in business. Again.
But dude, seriously?