YN 568EXIII flash underexposing at 1/16 and higher outputs

You've got a mode mismatch between your pop-up flash settings and your 568EX III's slave mode. You either need to take your pop-up flash out of Commander mode, and use it as a regular pop-up flash in M to use the 568EX III in S1 (or TTL to use the 568EX III in S2). Or, you need to set the 586EX III into Sn mode (with the MODE button) to make it a "smart" optical CLS/AWL receiver.

"Commander" modes on a Nikon body are for the Nikon "smart" optical CLS (Creative Lighting System)/AWL (Advanced Wireless Lighting) triggering. This gives you TTL, HSS/FP, and nearly full remote control over the off-camera flash. But the settings are communicated to the remote flash using light bursts, or pre-flashes. Kind of like morse code with light, or how a tv remote communicates with the TV.

The S1/S2 slave modes on your YN-568EX III, however, do not use this system of communication. They are "dumb" optical slave modes, where a sensor, when a flash burst is sensed, fires the flash, and that's all the control over the remote flash you have. S1 fires it on the first burst sensed, S2 fires it on the second burst sensed. S2 exists in case your pop-up flash is in TTL and uses a metering pre-burst.

What's happening is that the communication pre-bursts for CLS are tripping your flash in S1 mode too early. At the lower power settings, the flash can recycle quickly enough to fire again on the main burst, but at higher power settings it takes longer to fill up the capacitor from the batteries, and it can't keep up and fire in time, so you get a dark frame.