Thanks for the replies. I worked on the checkering for a couple hours today.
I decided to go for the "Both" option.
I used a steam iron according to
Patrick Sweeney's recommendations*:
Soak a cloth in water, and let it sit on the area to be "lifted".
Apply the iron, and let it sit as long as possible, without completely drying the cloth. (Much longer times are possible with a steam iron, so I didn't let it sit very long, at first.)
Repeat, as necessary, until the desired result is achieved.
Because of the odd pattern of steam ports on the iron I used, I had to get creative with positioning, and keep moving the ports to a different location on the stock with each repeat of the step.
In the end, the previously "negative" (low) spots in the checkering had been lifted enough to be significantly "positive" (high spots). The best side of the fore end had the inverted checkering lifted almost 0.050" above the surface, including the paint that had previously been stuck in the holes. It is really quite odd to see "negative image" checkering on a stock.
After the stock dried thoroughly, I sanded the lifted portions until flush with the stock surface.
It looks like I will need to hit all three areas again (left, right, and bottom), but the progress thus far is looking good. There won't be much more wood to be removed, should I decide to completely eliminate the checkering; and if I decide to re-cut the checkering, there will still enough wood for that, as well.
If anyone was wondering - the steam process didn't loosen the paint at all. It was still well bonded to the lifted wood (and is the reason some areas didn't lift well - and will be needing a second attempt after a little detail work).
*Sweeney's recommendations are for a stock iron... but I use what I've got, and the steam seemed to help quite a bit.
You might try some lacquer thinner, and if that is not working MEK (Methyl Ethyl Ketone).
Previously tried on this stock and/or action (under various brand and product names):
MEK
Lacquer thinner
Carb cleaner
Brake cleaner
Brake fluid
Mineral spirits
Citrus Brite (may be misspelled)
Purple Power (wicked, wicked engine degreaser and paint stripper)
Turpentine
Naptha
Kerosene
Denatured Alcohol (I hate this stuff.... It didn't last long, even though it worked on some "contaminants")
Some kind of biodegradable, "earth friendly", universal "Paint Stripper". (it was crap!)
Not all of the above were used specifically for paint, but were tested on it, if the opportunity was there. (This was one dirty, nasty, grimy, paint-encrusted rifle; with coats of a variety of paint types.) Some of the chemicals were eliminated for vapor/odor concerns, and some were eliminated because I hate working with them. One triumphed over all.
Acetone was the big winner. It took all of the upper coats off fairly well, and was the only thing that worked on the base coat in a timely manner (some kind of enamel, I think).
I'll update this thread, as the stock progresses.