Yet Sisko couldn't do it himself. As Garak pointed out, he _had_ to go to Garak for help from the beginning, knowing, conciously or not, that he would never be able to go through with it by himself, that he had to have someone like Garak, not just to obtain the data rod and manufacture the holographic record, but someone who could bypass any of Sisko's moral obligations and do what was needed, that is, not just trick the Romulans, but kill to pull it off, something Sisko couldn't have done (though he can kill in battle--though they all can, when it comes down to it...except maybe Chief O'Brien, who was left feeling so awful after the first time he picked up a phaser and killed an attacker way back before Deep Space Nine, before the Enterprise).
As to whether Kirk or Picard (or Janeway would have done it, we haven't seen any of the three put in such situations as Sisko was in. He was CO of the one thing standing at the gate to the Alpha Quadrant and had been in the center of a long, actually-declared war against an enemy that cloned its soldiers and change its own appearance. Friends and comrades were dying right and left. Although supposedly we were seeing Picard in the same political situation, we never saw him and the Enterprise actually involved in the Dominion War. We've seen them all being pretty tough when it comes to protecting their crew (think of Janeway in Equinox); it's possible that when pushed, for those they love and feel a duty towards, they would go as far as Sisko did in IPM.