AIFF is decoded by a DirectShow filter. (By default MC uses Microsoft's standard Quartz.dll filter.)
The exclusive WASAPI mode doesn't allow the DirectShow filters to put audio through.
You could convert the AIFF files to Wave or a lossless format (APE, FLAC or WMA). They would be decoded through MC's internal playback engine and would work with the exclusive mode. Though, probably you have a good reason to use AIFF instead of the more common PC formats.