Although Soundbooth presents many regressions compared to Audition (aka Cool Edit pro !), the main problem seems to be the documentation. Copying one chanel over the other is indeed very simple:
1. Export yuur file to separate channels (File > Export > Chanels to mono files).
(Please note that this is a translation from the French version of Soundbooth.)
You should get two files with the same name as the original an “_L” and “_R” sufixes.)
2. Open the file corresponding to channel you need and use “Save as”. In the “Options” dialog box, select “Stereo” instead of “Mono”
This is a workaround to Tristan’s problem. Compared to Auditon, it is like copying 100% of R + 0% of L to L (or the opposite). Compared to what one could do with Auditon, it’s a regression, and not a small one!