On my opinion it can be done by profile cutting first the inner slots with tabs, avoid hatching it because it`s so time consuming and you will surely break a tool bit in the process, a 5mm pass will be just fine even if you just use a high speed steel end mill, the codes will be divided into 4 parts, inner profile cut, surface contour cut, inner profile cut on the drawers and drawer face profile cut, attacking it via 3D process will surely stress it`s either your machine or your tool bits used. I can help you with the codes if you want, I have worked on furnitures bigger than your drawer. if you divide the work process it`s much more easier than machining the whole thing straight, even if you say it can fit on your CNC table.
Advantages on dividing the codes - you can assign a tool bit used in each code, you can re sharpen the tool bit and reload it on the next code as so on and so forth.