I don't think Martin used anything unusual in tuner-mounting screws. If they used a Grover tuner, then whatever Grover puts in the box is what they used to mount it. The little set-screw to hold Grover Rotomatics or Sta-tites in place is a very common screw.
If it was a 1930s Martin, or a "New York" Martin (pre-1896), then you might have a worry. But a 1960s Martin? No, they used nothing extraordinary for the purpose. Any good repairman can provide you with a Grover mounting screw. I was thinking it was an older model, somehow.
You have mental pictures from what you're familiar with and I didn't connect with my 1960s D-28, but instead with the Martin I'm playing a lot these days -- a pre-Civil War Martin model 2-27 (on long-term loan, as I couldn't afford something like that at all) and the tuners on that are nifty brass tuners with ivory pegs. Those tuner mounting screws might be hard to match, as they are tiny flat-head brass screws, not round-head nickle screws like modern tuners sport.
You're fine, whatever you do. Collectors of vintage instruments are not so picky about a tuner screw, or even a tuner as such. I have a 1944 Epiphone Zenith, which, when I got it, had three original tuners and a modern, 3-on-a-strip set on the other side of the peghead. i just bought a replacement set of Grover Sta-tites and put the remaining original tuners in a box. If ever I sell the guitar (or my estate does so), the originals can be put in a baggie in the case for the buyer. I don't think the difference in value would be much if it still had all original tuners. And even if the difference was significant (>$100, let's say), if one of the mounting screws was changed out, it wouldn't make 10 cents' difference.
Once that guitar is 150 years old, the mismatched tuner screw might make $10 difference in the value, but by then, $10 will be a postage stamp's cost.
Get a new, modern screw, and play the guitar. It's not doing you any good without the tuner.