Before you start, bear in mind that cleaning these areas is cosmetic only. Build-up on the bars and under the screen do not affect the operation of the Box. Like with most grinders, there is an initial buildup in some areas which eventually stabilizes, and then it becomes just "part of the operation" -- material is neither lost nor gained. Magic-Flight generally recommends that people leave them be and concentrate on maintaining the aesthetics of the exterior wood. If you still feel compelled to clean, proceed at your peril.
First, the bars: heat does help to soften the residue so insert a battery for 10-15 seconds with an open cover and empty trench. Soak the brush in isopropyl alcohol and then brush the bars clear while hot, but make sure to keep the brush bristles at least damp with alcohol while doing this. Note that the mesh is bent over the bars, therefore the area you are brushing is probably the weakest part of a delicate screen.
Under the screen: scraping is really the only option, but a small slip can ruin your screen. Cleaning under the screen is neither necessary nor advisable, so there are no recommended tools or techniques.