I lap with a DMT Coarse Continuous Diamond hone. I do about 10 coticules a week and have been doing that for several years. The Coarse stone doesn't clog up and I lap in the kitchen sink under warm running water to wash away the swarf. It's counterintuitive but a Coarse 325 grit stone leaves an 8000 grit coticule purring like velvet. It's quick too! I lap flat in under a minute and that goes for vintage coticules from eBay as well which are almost always dished significantly.

Don't worry about the color in the stone as long as the stone is lapped flat. Hold a straightedge onto the surface of the stone lengthwise with the straightedge on edge. If you can see light between the straightedge and the stone, you've got more work to do. I use a 6" Starrett Machinist's Rule as a straightedge and I'll send you one for $3 if you'd like (plus shipping) which should be about $1 as I could put it in an envelope.

Starrett tools are the most precise in the world and they're made in Athol, MA. I use a lot of their tools. "Flat" is a specification and not a state.