Nah I wouldn't use the big dive knife for tree work. The small one maybe, but the big one is hugely overkill with like a 9" blade. Heck, it's overkill for diving!
The Stop is for use on single ropes from 9 to 12mm. That's the same as .35" to .47" which falls below the size of your 1/2" rope. So, the reason you'd be having trouble with the half inch is becuase the Stop WASNT DESIGNED for 1/2" rope. Ya might bother to check the specs of the pieces you purchase before ya get em
I imagine 7/16" would go fairly nicely through it.
Also, the double-cam on the other is possibly a panic over-caming brake. Some of these that you use a lever to control the rate of descent have those. People start falling, freak out and pull the handle harder as a reflex motion, when in reality that only makes them fall faster. So they installed an additional deal in there that starts jamming up if you over-cam it.