Unfortunately I can't actually get to the top of the crack without removing the scuff sheet in one way or another as the entire crack is covered by it. Think it's going to be easiest to cut it around the crack and bend it to one side so the whole crack can be glued.
The easiest thing to do is remove the scuff sheet from edge to edge to reduce willow strip, carry out the repair and then sand the entire face clean, then apply another scuff sheet.
If you don't have the equipment/confidence to sand the face, or you don't have another scuff sheet, then use a stanley knife to cut the scuff sheet just above the damage across the face at 90 degrees to the edges, then peel away the section of sheet below the cut, exposing the damage. After repairing, apply fibre tape to the exposed area.
PVA glue is in my opinion the best glue, it's certainly the most versatile. There is no need to water the glue down in my opinion, it will only serve to weaken the bond. If you are concerned about filling all the gaps in a crack, you can inject the glue using a needle (not necessary in this case in my opinion) or use a straw to blow the glue right into the gaps. Tightly wrap with parcel tape and leave the bat in a position where gravity will continue to pull the glue into the crack.