I've been following this with great interest but a little confused as usual.
You seem to be getting advice about photographing the bear, not kids in motion. With any of the shots you have posted, the bear or the kids, will pick up different light intensity in each spot they move to because you are unable to light the backgrounds and floor evenly with your four lights.
Your are trying to photograph children 'running around and playing', so you won't be carefully aiming light at them individually, they will have to be moving about within a 'perfectly lit' space as opposed being perfectly lit themselves in a space. I can't see how that can be done, they will be moving through any light aimed at any surface or reflector, which is uneven at best.
Seems to me you would be better to try and create a sort of giant shadow-less area sort of like a giant light tent with light coming through the back and sides, thrown back down from the top with reflection onto the floor.
Once that bits done, it's done, like a sort of giant light tent type of special set. You could then do whatever you want with light from the front aimed into this space to pick out the kids that are then moving 'through' perfect lighting as opposed to being randomly lit in a an unevenly lit area.
Is that arse about face or am I blundering onto the right track.