It won't hurt him at all

The way I used to do it was to wait until bubs was asleep, then drip a bit into the corner of his eye, so it kinda pooled in the inner corner. Then I'd move the baby very slightly, so it kinda moved into the eye. When he would wake up, the gunk would be completely gone, it is magical wonderful stuff, that breastmilk!

Another way is to express and use a cotton ball. Dip it in, wipe from inside to out and only once.

I prefer the squirting-in-the-eye method, myself