Along with broken handles this is the most common type of damage to bats anyway
now you have established its an hairline crack and goes all the way through the bat here's what I would do dont force open the crack place the bat horizontal and apply the glue around all accessible cracked areas and bind tight with package tape when the glue is dry remove the tape and gently sand off the excess glue using 240 grit sandpaper apply scuff sheet and tape the damaged area tightly with fibreglass bat tape. Then use the bat as normal as the bat could last for ages. Good Luck.
I agree with the diagnosis but not the method.
Personally I'd do it in two or three three stages.
Lay it face up and apply glue to the crack. I would personally use the glue as it is for this repair, as watering it down is a spin off between more "seepage" and strength of the bond.
Laying the bat so the area glue is applied to is flat will allow gravity to do its thing and pull it the glue down into the crack. Don't clamp or tape it, if it's a hairline crack you want to let the glue get in there, not seal the crack so it can't.
You could also use a drinking straw to try and blow the glue further into the crack, but this isn't an essential step.
Once the glue is dry repeat the process with the toe and back being the "flat" area. This will mean that the glue has been able to seep/penetrate into the crack as deep as gravity will allow it to go.
Once all the crack has been done, and the glue is thoroughly dry, give it a light sand and apply some tape.