I don't think taping up, or using old grips is going to do the job for this one to be honest, I would really try and clamp it in a vice or with g clamps at least.
I would dowel it too, you only need to go about an inch or so past the line of the joint to suppot it, but going further probably won't hurt.
Glue the flap down first, using grips will be fine for that bit.
Then cut it all square, then cut the replacement bits, I would use 2 bits looking at it, better a bit on the big side than too smal, then glue them in, claping them across the bat - edge to edge I mean, the shape down roughly, then dowel, then sand to finish!