Drill = bad. Avoid if at all possible, it's hard to go back without a messy fix. If you *must* drill, use a bit that is a smaller gauge than the screw (obvious but easy mistake), and only drill as deep as necessary. Measure the depth of existing hole versus the length of the new screws, allowing for the washers and strap button, both of which will lessen the required depth. Then wrap some masking tape around the drill bit to mark the exact depth required so you can see how far you've drilled as you are doing it.
Alternatively just use a dimarzio clip lock strap. No drilling required!