[shoot summ]

You will need a counter bore bit. Has a pilot that guides the bit from the through hole.

Counter bore:

McMaster-Carr

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McMaster-Carr

McMaster-Carr is the complete source for your plant with over 595,000 products. 98% of products ordered ship from stock and deliver same or next day.

www.mcmaster.com

Longer screws would be cheaper and easier.

 

[shoot summ]

A same size Forstner bit would "self-center" in the existing hole, more common and cheaper if not already owned by the OP.
It can NOT be a saw tooth type bit, must be flush cutting on the bottom.

8mm(5/16") is not a common size in Forstner sets, who knows if the OP even has a typical set, let alone an atypical size?

A 5/16" straight cut router bit is an option as well.

All of this is assuming the OP has a drill press.

 

[shoot summ]

I thinks if the OP had a drill press, he wouldn't have had to ask.

I thinks we really have no idea if he does or doesn't, people ask a lot of questions, some have equipment, some don't, the question in no way is a clue to what they have.

Suggesting cutters that require a drill press would require the OP to have one, or access to one.

Buying longer screws wouldn't require any of that, but would require getting the right length, too long and they can interfere with the mag.

Regardless it looks like the OP is headed to longer screws.

 

[shoot summ]

Well.. I called the manufacturer to ask for the screw specs, and found out they sell longer screws for wood grips. I had no idea. So thanks again for the suggestion. Glad I didn't spend money on tools to modify the holes and possibly mess up the grips!

Glad you figured it out, thanks for coming back and telling us the solution.

 

[shoot summ]

Length and thread size are only two of the requirements for a firearm. The hardware store screws will be either cadmium plated or possibly stainless, not blued as would be most likely on a quality firearm.

Not to mention the low profile, typically larger diameter heads.
I've trolled the hardware aisle a lot, I would say there is low odds chance of finding acceptable grip screws there.

 

[shoot summ]

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.

You likely didn't read the whole thread, the OP contacted the manufacturer of the pistol, they actually produce longer grip screws for the wood grips, his issue is solved.