Thanks again for your input Eric.
Looking back, and reading what you've written, as good as my intentions were I handled this fish more clumsily (and selfishly because I very much wanted a photo) than I should have. But I will get better at it and try harder to do what's best to keep these fish in good shape.
When I've been fishing for smallmouth, I have clamped down the barbs on the flies. Honestly with these tiny trout hooks it didn't occur to me to bother with the barbs. But I'm releasing everything, so I will start doing that too. By the way, the other two trout I caught spit the hook out pretty much as they were getting in the net, so this is the only fly I've had to remove.
What about holding the fish through the net? Like, get him into the net, then hold onto him using the fabric of the net (the one I'm using is called a catch and release net). Might be a good way to get a grip and control him, while not handling him directly, long enough to remove the fly?
Anyway, thanks again.