I've never used a baker's couche (and in fact had never even heard of this tool before reading this question), but I do know a little about fabrics, and it seems that a baker's couche is nothing more complicated or mysterious than a piece of heavy linen fabric. Based on that, I'd say you'd be perfectly fine to wash it.
Linen is unlike most other fabrics in one important respect: it actually gets stronger when wet. This means it is perfectly 100% washable, despite what clothing labels try to tell you. The reason linen clothing is traditionally labeled "dry clean only" is that linen shrinks when washed, and clothing manufacturers don't always pre-shrink their fabrics.
Besides shrinkage, two other things happen when linen is washed: it gets wrinkled (well, OK, so it gets wrinkled if you look at it cross-eyed, but anyway), and it gets stiff. To remedy both of these, you need to iron it dry. As in, start with wet (not just sprayed-with-a-spritzer-bottle barely-damp-on-the-surface, but actually wet) fabric fresh out of the wringer, and the highest setting of your dry (as in, steam turned off) iron. (There's a reason said highest setting is often labeled "linen".)
Now you'll have linen that is not wrinkled, but is probably a little shiny and crisp. With clothing, five minutes of wearing (especially when it's hot and humid, i.e. perfect linen weather) will solve that. For a couche, what you may need to do is loosely roll it up, put it on a sturdy wooden surface, and whack at it with a rolling pin or clean stick. Or you can just start using it - it should soften right back up with use.
As far as whether to use detergent and what kind, since this will be used next to food, you want something that will rinse out thoroughly; but since you will be ironing the fabric, you also don't want any food residue in it to get all scorched and icky. Personally, I'd use a delicate-fabrics detergent and lots of rinsing in warm water.