To both check and clean checkering, I use the thread guage, that comes with most tap and die sets. I have even used one to recut over the filled spot, when I've had to fill in a chipped or gouged spot with wood putty.
In the spirit of keeping things simple, it's fairly rare that checkering actually needs to be recut. Usually a freshening will do a 100% job.
Go to Brownells. I use Gunline tools, so I'll reference them, and if you decide on Dembart, that's okay because they'll both do a good job for you.
What you want is a 90 degree pointer in a medium cut. This is a single-line tool. It's more a file than a cutter. It works quickly and leaves a nice surface. Minimally you should have two handles and cutters, a long cutter (PL90) that works on a push stroke, and for tight spots a short cutter (PS90) that will cut with a pull stroke.
"I do not want to cut brand new checkering, only clean out the existing checkering pattern."
The lpi doesn't matter. To refresh checkering all you need is a single line cutter. Since you mostly just want to clean out varnish and remove as little wood as possible (zero is good!), I suggest a 90 degree, single line, FINE cutter. Put it in the handle to cut on the pull. You can use a back and forth motion in the middle of the pattern, but always stop within 1/4" of the border. To do that 1/4" nearest the border, lift your cutter, place the point carefully at the border and pull the cutter toward the center of the pattern. Overruns are caused by charging the border and cutting on the push makes it very likely.
The more lightly you cut the better. Imagine you're going to have to go over it 5 or 6 times to get it done and you'll be working at the right pace.
If you want to do a good job on recutting checkering get both a 60 degree cutter and a 90 degree cutter. Go over the checkering with the 60 degree first to clean the grooves out and bring them to the proper depth. It will be easier to keep the 60 degree in the individual grooves. Finish up with a light cut with a fine 90 degree cutter and seal with a very light coat of TruOil applied with a tooth brush.