Polishing should never groove a lap. Either the facets are not flat on the lap, or you are using far too much pressure. It’s also not the fault of the Diastik. Diastiks are self-lubricating and lubricant is only needed to spread the diamond paste evenly. I squirt a bit of WD-40 onto a piece of paper towel and wipe the lap moist with that, then apply the Diastik lightly on a slowly spinning lap. You can spread the paste more evenly with a clean finger. If the lap needs lubricating I apply a very short spray of WD-40 near the center of the lap – the tiniest squirt possible. The polishing lap should be damp, not wet.
You can resurface a damaged Batt lap, or any other soft metal lap, on good quality 80 or 100 grit wet-and-dry paper, held down by water on a hard, flat surface, like a sheet of glass. (I use Klingspor automotive paper on a polished granite tile to resurrect second-hand laps.) Rub the lap by hand in a figure of 8, rinsing the paper with water when necessary to remove clogging metal. When it is flat, give the lap a good wash to remove any abrasive grains. The resulting random texture of scratches on the lap is fine but if you want to you can finish the lap on 600 grit paper.