Ok, this seems to be a lesson in not always trusting your host's tech support. They assured me that their built in web caching tool was just another level of enhancement, and that it wouldn't interfere with any plugin cache. In fact, they pretty much said that they couldn't do any work to support my speed issues if I turned it off.
Anyway, I went ahead and turned it off, and..... wait for it...... Now both sites are running W3 Total Cache perfectly!
I should really close this topic, I know, but can anyone tell me if I really need a .htaccess file on my root at all?
