One of the dangling questions left over from "The Enterprise Incident" was how necessary was it for the Federation to risk the Enterprise being captured in order to steal a Romulan cloaking device in the first place. Did the newest revision to the Romulan stealth technology pose that great a threat to Federation interests?
In "Balance of Terror", it was revealed that a Romulan ship, so equipped, could destroy outposts along the neutral zone. But that's "along the border", in "no man's land", if I read that correctly, and the Romulan suffered from fuel exhaustion from the repeated use of the cloak and the plasma weapon. It seemed doubtful to me that a Rumulan ship, or ships, could penetrate deeply into Federation space with that handicap. And while cloaked, the Romulan ship had trouble tracking the Enterprise because of the cloak.
The only thing we learn for sure from "The Enterprise Incident" is that the newer cloak is much harder to detect, "rendering our tracking sensors useless" according to Spock. That's a tactical, but not strategic advantage. There is no indication from the ep that the Romulan's other deficiencies have changed significantly. The only thing we see is that when the flagship flies by the evasive (and newly cloaked) Enterprise, is the Romulan flagship on the bridge's main viewscreen. That doesn't really tell us much. We don't know how much the new cloak will drain the Enterprise, how effectively it can hide a ship the size of the Enterprise, for how long it will hide the Enterprise, or what the side-effects would be to the Enterprise's tracking sensors while operating under cloak. It may be reasonable to expect that a Federation cruiser under cloak could only manage Warp 3 for limited periods, and not be able to go faster because of tracking "forward visibility" or the drain on the ship's power.
Could the use of Klingon-designed starships be the wild card here? Possibly. Combining the new cloak with Klingon ships of greater range, size and firepower could add more motivation to the Federation authorizing such a mission.
Clearly, the Federation could have authorized the theft of this device just for dissecting/analysis purposes. They may want to know how it works and what its limitations would be so they can devise a defense against it. They could also desire to crack the device's vulnerabilities so it would be easier to modify scanning systems to "track a vessel so equipped". That would seem to be a prudent strategy to pursue, if still a risky one.
But would the Federation engage in this kind of mission just to steal a rival power's weapon for its own sake, as a kind of Cold War game of "it"? I still have my doubts.
TNG's "The Pegasus" seemed to suggest that if the Federation wanted to employ cloaking technology they could develop their own and it might even be superior to that of the Romulans.