There are plastic plugs for badge fitting. The plug press fits into the holes and the plugs have a small hole through them for the legs of the badge. The legs should fit tight. Examples here. Look at 803 types. Use a bit of silicone for better security if they are a bit loose. Many suppliers of these. 1/4" sounds a bit big but if you've had the bonnet painted the painter might have skimmed with filler and made the holes rather big.
https://www.vintagecarparts.co.uk/pa...dex/panel-clip