Checkering a new stock is done after finishing. Therefore, I suggest refinishing to perfection, making an effort to keep the finish out of the checkering...only because it will have to be cleaned out and filling the lines to heavily will only make it more difficult to follow them. I would finish into the border or maybe 1/8" into the pattern itself, if it's borderless.
After recutting, the wood will be raw and completely open-pored. Just like the initial finishing of bare wood on a new stock, I like to saturate the checkering with my finish cut 50% with mineral spirits for maximum penetration and sealing with minimal build-up. I apply 2-3 coats, virtually never more than 4. My goal is to build the slightest sheen in the checkering (which tells me it's sealed) without build-up. It shouldn't be a lot shinier than the surrounding wood, nor do I like the light, dry look that tells me it was freshly done and which suggests it wasn't properly sealed. If you can drop a single drop of water in the checkering and leave a spot, it's not sealed, meaning gun and skin oil can penetrate the wood.
