Firstly I will say sticking your fingers in a fishes Gills and using that as your method is not good for the fish. Would you like a giant to grab you around the neck and suspend you to pose for a photo? That's their breathing apparatus. Not aimed at you directly, I'm just generally putting it out there.
Iv found if you're slow and deliberate, you can gently raise fish like bream, whiting, black drummer, fish of that general size, by cradling their under side and letting them sit in your hand, especially after a fight when they're a bit knackered. Don't try and grasp the fish between your thumb and fore finger, can gain a nasty deep gash in either like I did. I almost feel like Steve Irwin (when he was a alive) when I have a fish sitting obediently in my palm posing for a photo.
Fish like kings and salmon, same thing, support its underside, only difference is I use my other hand to grasp the tail.
I don't like handling flathead, I try and use a rag, iv been stabbed a few times by the spikes on the side of its head, if I do have to handle one, I let it settle half a minute after the fight,if I don't have lip grips, I put my thumb in its mouth and grasp the skin of the jaw between my thumb and fore finger, and support its weight in the other hand, might get a free scratches on your thumb, but nothing to bad if you don't clamp it's actual teeth.