The first measurement you need is the bat’s mass. Should be able to find that with a spring scale.

You can also determine the COM via rotational equilibrium by hanging the bat horizontally by a support at one end and a spring scale at the other.

I assume this is a fairly modern HS physics lab and a photogate is available. Ever use one?

The photogate is able to measure linear speed when a thin cardboard “strip” of known width passes through it.

The strip’s mass is negligible and can be attached to a position on the bat such that it passes through the gate at the bottom of its swing.

Finally, I’ll tell you that conservation of energy is probably the easiest way to determine the bat’s rotational inertia.

$mgh = \dfrac{1}{2}I \omega^2$