I know that someone tried this out a few months ago, again with no luck.
You're right, in theory all that should need to be done is add an entry in the mime types file, but that doesn't seem to work.
The FLAC codec works for both playing files directly and for playing from streams (for SlimServer/SlimRoku) so I wouldn't think theres any issues there, I think Sonata is just choosing not to use the mime types file.
I downloaded a trial version of TwonkyMedia and it played FLAC files perfectly for me!
It appears it was using the WAV codec to play the files as it was able to play some of my DTS music which was encoded in FLAC format (which wouldn't work if it was trying to transcode to MP3).
I did some tests to confirm this and TwonkyMedia is definitely sending content via the WAV codec.
Now the reason it works for me (I'm pretty sure) is that I have a modified WAV codec on my machine suitable for streaming (developed for use with SlimRoku).
So, if you want to try this out, heres a link to the suitable WAV codec
Hopefully this will work for you as well.