I just use wood filler for those sort of holes where they'll be on display and won't need to take a screw. Then I try and get the filler the same colour as the maple. There are products like this. https://www.amazon.co.uk/dp/B003OBX6...dDbGljaz10cnVl, though I normally use a spirit-based marker.

The holes will be smaller than 3mm diameter as that's the full screw diameter, so probably around 1.5mm-2mm.

The veneer idea works, but you then get a line along the edge of the headstock. A bit swings and roundabouts.

It's probably too late to suggest getting replacement tuners that use the same mounting holes.