The result was that - with the exception of pidgin, keepassx, and opera - all of the tray icons moved to the other instance of the notification-area-applet and did not seem to be otherwise affected. Then I removed the second instance of notification-area-applet. Then I launched the following (list includes aforementioned pidgin and thunderbird):

Pidgin and Thunderbird are set to start with my Gnome session, so both icons appeared when I logged in (though, of course, Pidgin's window appeared on screen too, so I minimized it). I re-added the second instance of notification-area-applet to the new panel and rebooted. I thought maybe Thunderbird is just an exception, since I am using a plugin to get that functionality in the first place, so I decided to do more testing. However, the buddy list window for Pidgin re-appeared! Thunderbird's window did not. I've found that if I remove the notification-area-applet from the new panel, the icons (Pidgin and Thunderbird) moved over to the other instance of the notification-area-applet. When I boot, it seems that whichever instance of notification-area-applet loads first is the one that actually gets used (when launching applications after booting, too), but that isn't the point I have to make.

I've done some testing by adding a new panel with a second instance of notification-area-applet.

Skype minimize to tray