I drilled out the plugs today - they were timber dowels. I don't know what they had been glued in with... whatever it was had dried hard and white, but evidently didn't bond very well to timber.

I took Scarecrow's advice put some fibreglass with epoxy on the insides, then filled the holes with epoxy mixed with micro balloons.

Next week I will go back and sand off. I might also put a little bit of glass over the filled holes on the outside, for good measure.