For lapping, I always use wet/dry sandpaper. I dont use tape to hold it down as a little water will cause plenty of "stiction". Sandpaper is cheap, DMT's aren't. I usually start with 400 grit & stop at 1K, with nothing in-between. I was fortunate enough to get a piece of granite that was a "cut-out" for a kitchen sink for free. Before that, the counter-top worked just fine...just had to pay particular attention to not scratch it...SWMBO kept her eye on me.
Back on the C12K/PHIG, I wasn't fortunate enough to find one that would cut fast enough for me. I'm sure this is why I don't personally recommend them. I bought two hoping to get one that was worthwhile, but it wasn't in the cards/my luck. The big one is HUGE & you could easily use the side for those difficult blades that may have a bad grind.