I mean swapping to something better always an option, but those seem like custom-made, hence I assume OP wanted those specific grips, so don't think that's an option.yomanbill said:
The checkreing in the bottom of the counter bore is an interesting touch. I think that before modifying the hole I would try to find some longer screws. If that doesn't work, get a forstner bit of the proper size and set it up on a drill press. I'm assuming the screw hole is perpendicular to the surface of the back of the grip. If not, you will have to shim it on the drill press so the counter bore will remain in proper alignment. Of course, if you do it this way, you will lose the checkeing in the counter bore.
Further thinking; could you use a small chisel and carefully shave off the tops of the checkering at the bottom of the hole. Would that be enough?
The checkering in the screwhead relief is probably do to the checkering being impressed first and the relief hole was formed by subsequent impression forming.
My suggestion it to return these inexpensive Chinese grip scales and buy a higher grade replacements( ideally you would find some local gun store like Gun Store | Guns For Sale Online and check what they've got, but in my experience you rarely can get quality wooden grips there, it's all polymer/polytape usually).