So thanks to David Satter for the link to 3/16" brass grommets, and to Fitz for the pic of a tool used to shape the outer rim squish flat washers.
The pic above shows what I'm trying to replicate. These 'cup washers' conform to the cone shaped outer sides of the head of a #8 flat head screw. They have a flattened lip at their outer edge that rests against the top of the wood. They serve to effectively increase the outer diameter of the screw's head thereby making it less likely the screw will bury itself into the wood. You'll find there are different sizes and shapes out there depending on the canoe manufacturer's whim. Some have just a cone shape; others also have a flat outer rim that also helps to keep the washer from digging into the wood. They are typically used with keel screws, although you'll sometimes also see them used with screws holding the decks in place.
Here's the grommets I bought. Got like 100 for about $10 with shipping as I recall. Each grommet is made of two parts: the base which has a wider outer edge and post, and a thinner washer. When you squish the two together the 'post' deforms to hold the washer. As they are not ideal as the outer edge will dig into the wood.
Happily I have a retired machinist buddy who is just aching to make stuff for me. Using Fitz's idea he made this 2-part fixture. You simply place the grommet (either the pert with the pose or the washer) onto the fixture with the pin, then place the upper part on the pin so it's against the grommet, then wack it with a hammer. This flattens the outer rim of the grommet. Works great.
The washer part of the grommet should work in all situations. The grommet part with the post will also work if the hole in the wood is wide enough.
And here's what they look like. Note that the screw head is essentially flush to the wood and not sticking way up as with the typical 'finish washer' you see sold today.