I did ask a similar question in another post based on the Logitech Pure HiFi anywhere and the Intempo iDab.
In theory you can slot the iDab in the docking port of the Logitech to give you DAB.
You can take the whole lot with you and because it uses a rechargeable battery is ideal for the bathroom where you don't want to be involved with mains eleckie (unless you are known as vanman)
Downside is you have to use headphones for an the aeriel and not having found any reviews of the product was hoping for some reply from WHF on the iDab.
My experinece of DAB (living on the sticks) is that you need an extrenal aerial so you may need to look at a wirefi enabled Internet radio with docking station and stream accordingly.