Thanks to everyone who replied.
Before I opened this thread, I had bought one I thought would work (top of the image below). On the first fish, a mid sized jacksmelt, the fish wiggled enough for it to break through the membrane of the lower jaw. I figured the contact area was too small, and I haven't tried it again since. I would be concerned about the same problem with the mini fish gripper posted by pinchepesca.
I found the FISH GRIP "JR" at a local tackle shop. It worked for barred surf perch, but I was concerned that it was too big for smaller mouthed fish, so I carved the jaws down a bit with a Dremel. I hope I have a good compromise in the size of the jaws. I will try it on the next trip.