I found the culprit — in app/code/community/IllApps/Shipsync/Model/Shipping/Carrier/Fedex/Rate
GroupPackageCount should be 1 (or, better stated, should have some better logic for handling values > 1). Line 417:
$request['RequestedShipment']['RequestedPackageLineItems'][$i]['GroupPackageCount'] = 1;
Basically, we're looping over all the packages, then saying each package (1, 2, 3..) has n number of packages in it (1, 2, 3).
It takes care of the error message ("Requested shipment package count did not match the summed total of group package count values; the summed total of group package count values was used to rate.") and makes the rates much more sensible for multiple-piece shipments.
Hooray!
Jon