From their FAQ:
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Which media players does your service work with?
Our service works with Winamp (Windows), XMMS, Apple iTunes (Mac/Windows), Real Player and Windows Media Player.
For Palm OS: Pocket Tunes (deluxe version, after 1/15/2004), and many other players.
Which media players are not supported yet?
So far we have reports that MusicMatch player does not work correctly (use Winamp instead).
There may be others that do not support "basic authentication" correctly. Some hardware home audio setups may not support basic authentication correctly.
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So your request is for MusicBee to support authentication.
Have you tried simply appending the url with your login info? Eg.
http://listen.di.fm/private/example.pls?user=YourUsername?pass=YourPassword