MIMF

Thanks Bryan, it's good advice. And reasonable for anyone else in the future looking to solve a problem similar to mine. However, both the two big box stores within a reasonable distance from me have had their samples so torn up that last time I tried to use their gauges, I couldn't get the nut to thread on.
If I happen to head down to one in the next week, I'll be sure to take the post along with me. But I'll likely just see what arrives in the mail early next week. I'm hopeful it will fit. The dimensions look good, I think the only question will be thread pitch.
I appreciate you all being so helpful, and continuing to share your good ideas. Over such a small thing. Wonderful community.

Success!
The mystery can now finally be laid to rest. The #4-40 x 5/16" screws from Bolt Depot worked a treat.
The guitar is looking great with the new pick guard and tuning machine repaired.

Thanks everyone for your hospitality and help.