I've never had a BSB not be able to swim away and back down to the bottom, this one included.
I have caught quite a few of them over the years on the kayak so you need to learn how to handle them. That urgency is dictated by the fishes condition as it reaches the surface - some come up with liveliness and some are belly up and motionless.
-I don't puncture any part of the fish, and have never found it necessary. I wouldn't attach anything else to the fish that may cause it to tangle, or become hooked again.
Once I remove the hook, I peddle forward with my Hobie while holding the fish by the open mouth or by the gill plate. When water flows through the gills the fish will quickly recover and splash away.
(I can employ the same technique from the skiff)
That is exactly what was taking place in this pic that was shot by a fellow kayak angler.
Another successful release with hook removal.
With my method
You See a Live Fish Strongly Swim Away - in the Video I saw a floundering fish get dragged to the bottom with 12 lbs of lead after much handling.