Sounds like a practicable solution, with the punched 'plugs' glued to the screw-heads. (This is not what you'd normally do with a boat, and assumes you'll not ever want to withdraw the screws.) If the screws are below the bottom surface of the veneer, then you could build up the punched plugs in layers till you're above the surface and then sand back.
I guess I assumed you would remove the screw first & re-install it after patching & sanding-but whatever works for your situation.
You may have to put 2 layers of veneer plug to get above existing surface but maybe not if you punch lightly.
Use a good waterproof glue.
You could make your own punch from a sharpened piece of steel pipe.
