The cloaking bit was a late entry only provided to the Federation after the discovery of the wormhole. Sisko probably was asking for a ship since the first incursion with the Cardassians but Starfleet was spread thin. After the entire shenanigans with Odessy being destroyed by the Jem’hadar and the very real threat the Dominion was proving to be, Sisko probably threw a fit at being told there were no ships available to be stationed at DS9 that could carry out the mission. To which he probably pointed at the Defiant he knew to be in prototype phase and probably mothballed in a drydock.

At some point in the conversation the Romulan deal to supply a cloaking ship for use in the Gamma Quadrant in exchange for all the information gathered using it was probably made known to Sisko and he would have pointed out the Defiant was the only ship powerful enough to utilize it given the oversized reactor. So in universe it’s all just a happy accident they sent the one guy to look over the wormhole who had first hand knowledge about an unused prototype ship and vague theories on how to get it working on top of figuring out how to get a favor out of the Romulans in a way that pleased both sides.