If it was that simple, they could have done it with ordinary
cloaking devices, or at least made an attempt to do so.
On the other hand, they probably DID try this a number of times. Certainly the Maquis' implicit threat of cloaked interstellar missiles was credible enough for Starfleet to yank Eddington out of jail, which I suppose was the whole point of sending that fake message in the first place. If the Maquis could do it, the Klingons could almost certainly do it better.