Both of these options have positives and negatives, however I’m looking for other opinions on which to go with.
As far as I’m aware, a lighter flywheel will have a faster speed up time, while a heavy one will have more inertia(correct me if I’m wrong). This means that for a lighter flywheel compression would be much more important(again correct me if I’m wrong), but would be able to “double shot” much easier. While a heavier flywheel, compression is still important, but can power through it due to the inertia, but lose a lot of the speed after shootingedit: but will not slow down as much due to the inertia, however will have a slower speed up time.

Just looking to start a conversation on which you believe is better and why.

You are slightly off on your logic, a heavier flywheel will not slow down as much because it’s inertia will keep it moving, where a lighter flywheel will slow down more, but be able to spin up faster.

We have two 4" traction wheels and it seems to be a decent median, if we shoot two balls in a row with no slowdown they both reach the top flag. It also slows down to 0 in about 2s with a ratchet

If your flywheel is heavier then it will slow down less because of inertia , and if your only spinning your flywheel once in a match that´s only about 3 seconds used with about a .5 sec recovery time after shots, as where if your flywheel is lighter it will it only take about 2 seconds to first spin up, however the recovery time would be about 1.5 sec because the flywheel will lose much more inertia . Now this equals out but hey why do pitching machines have a heavily weighted rubber wheel?

Will you be running the flywheel the whole match? If not, you’ll be better off with the quick spinup of the lighter flywheel. The heavier flywheel will be good for running all match, especially if you strategize to wait until after the first 10 or so seconds to start shooting.

A flywheel is an energy storage device. If you can spin it, bigger is better. Run the raw numbers through a flywheel calculator(mass, diameter, rpm). A pair of 5” wheels, if you can get them, will deliver 3x energy of a pair of 4” wheels. We have ours on a ratchet and it takes 10 seconds to spin down.

Assuming the energy transferred is 100%, which isn’t true in the slightest. The energy required to be transferred is a constant in this equation, because everyone needs to shoot the same ball the same distance. Theoretically, a flywheel with the minimum amount of weight and radius would be the best to get the ball to the flag. Your motors have an easier time spinning up to speed, and you can be more efficient with your motors. Putting the minimum amount of energy into a system to achieve your goal will always be more efficient, so the smallest lightest flywheel usable will be more efficient, assuming the energy transfer is close the same percent efficiency.

The ball is definitely a constant but the flag stiffness and angle of the shot vary and more power is welcome in that case. Ideally, we’d like enough power to richocet the ball back into the intake. That’s why some folk drive Ram 3500 4x4 trucks and others Prius hybrids. Both get you where you are going but the Ram will more often than not overcome more variables.

While you're not wrong, I was assuming those were constants, as we're analyzing the differences between 2 styles of flywheel, and the same situation is needed to test the performances of both. I was simply saying the minimum needed, with the same level of efficiency would be better, because of the lower power required to start up, for example, a too heavy flywheel will never spin up, your motors just need more torque to perform. This would also lead to motors overheating more quickly, and more power drawn.