I need a clarification.........................are you looking to strip the old finish out of the checkering or are you trying to exclude the checkering from being stripped?
If you want to exclude it from striping, Mutinousdog has you covered.
If you want to get all the finish out of the checkering, use your Stipper on that just like the rest of the stock, let it sit a little longer and try to work it out of the grooves with a fine steel wool - like 000 or 0000. If that don't get it all out, then you need something to work into the grooves with the stripper. Ideally a single cut checkering tool would work great and they aren't very expensive at all - but you can probably just use a very thin bladed screwdriver to work the stripper into the grooves good, then wipe/rinse off the stripper according to it's directions.