If the laminate board can be lifted slightly more the hole can be drilled diagonally starting at the edge and may hidden when fitted flush again.
Consider injecting the glue using a syringe without a needle (medical or printer ink replacement from ebay). Some syringes have fairly long plastic outlet nozzles over which the needle is attached and would allow a hole of a few mm to be pumped full of glue. Alternative, if the glue consistency allows, some of the ink refill syringe needles may be of a wider bore to allow pumping glue down a 1 or 2 mm hole.