I've ground a fair amount of hardened D-2 though usually not as thin as your example. The wheel you're using should be a good enough general purpose wheel but would very much agree that a more "open pore" wheel should grind cooler. Also agree that your dressing technique/speed will affect the performance. Also agree that the total amount of material to grind off makes a difference in how many times the wheel needs to be dressed.
DOC you're taking shouldn't be a problem if the wheel is dressed correctly for the material. If you can't grind off .01mm (.0004) per pass then your wheel isn't correctly dressed (approx. 2-3 seconds for 13mm width wheel) or you have a very dull diamond. Have you inspected your diamond under magnification? If it looks rounded it needs to be replaced by a new diamond (or rotated in the holder, try that first) so there's a sharp edge to the diamond. A dull diamond will not fracture the wheel structure properly, it will only round off the grain edges. The burning you speak of is when the wheel surface is glazed and is just skidding across the material surface (leaving burn "skid marks"), not "biting" into the material and abrading it off as it should. I don't feel you need the CNC grinder for this but using coolant, even in a spray bottle, will help control heat/distortion/warping. Try creating a small "tub" from modeling clay around the piece and keep it topped up with coolant from the spray bottle.
Another thing that helped quite a bit with grinding hardened D-2 was the use of a spindle speed control (VFD) that slowed the wheel to 60-75% of full RPM. By slowing the spindle speed it increased the grinding force and caused the wheel grains to fracture, thereby exposing fresh sharp edges, where full spindle speed would cause the edges to dull. In effect it made the wheel "softer" and kept it from glazing/burning as readily. You may not have a spindle speed control available (ask for one) but following the suggestions we've all made should help considerably. If not then something isn't right and needs to be addressed. You should at least be able to get a new diamond from the budget if nothing else. Dressing should only be about .03-05 mm at a time, more than that is not good for the diamond, especially when doing a fast dress.
One more thing, D-2 is a more abrasion resistant material and generates more heat during grinding than most other tool steels. You have to keep a constant check on how hot the workpiece is and stop to cool it down or allow it to cool down. I've had jobs grinding blade punches that required me to take no more than 2-3 passes at .01 mm DOC and then allow the material to cool down or it would start to warp/deform/burn. After cooling off, (too hot to touch is too hot to grind) I could begin again. Your final pass to size should be with the material at room temp or cold to touch to ensure size/flatness within tolerance. I'm not the last word in grinding but a tool and die maker has probably had the most experience grinding hardened D-2 so hunt one up, buy him/her a cold one, and ask for advice. Hope this helps, good luck.