Actually, Klingons often transport when cloaked - Kirk used that ability extensively in ST4, and it played a role in "Unification" as well.
It's Romulans who insist on stopping everything else when cloaking, and conversely in dropping cloak when doing anything at all. This also affects the Defiant, with its cheap Romulan cloak.
That's true, which suddenly leads me to ask the question of why we don't see more Klingon ships trying to fire while cloaked.
Getting it to work is "new", and different from how regular transporters work, though.
It's new to Kirk and NuSpock. It's Trivia Night for SpockPrime, however.