If that "reason" was true the Federation would not only be asked, but it would be demanded, that the Cloak be removed from the Galaxy-X in THIS time-line.
It's amazing that people have not learned yet that it's not the cloaking that makes people good. All that giving the Federation any amount of cloaking will do is to cause the PvP scene to become incredibly boring.
Canon arguments are irrelevant at this point (neither side will agree on it), but the end result of what would happen is important, and Klingons know better than dedicated Federation players do about how boring KvK matches are.
Give it to escorts and it's a matter of time before it's in the hands of Cruisers and Science Ships.
No Federation ship has, or is allowed to have, a cloak. The only Federation ship to ever have a cloak was the first USS Defiant. That cloak was on loan from the Romulan government in exchange for data that Starfleet brought back through the wormhole from the Gamma Quadrant. When the Defiant was destroyed, Starfleet renamed the USS San Paulo to USS Defiant to honor the craft. However, the San Paulo DID NOT have a cloak and a cloaking device is not a part of the class design. No Federation escort class has a cloak as its standard design, not now and not ever. The Federation is not allowed to develop cloaking technology by the treaty of Algeron. The Galaxy-X DID have a cloak. However, that ship is from another universe. And because the Galaxy-X is from another universe, the treaty has not been broken. The Federation of our universe did NOT develop cloaking technology, therefore the treaty is intact.
THAT is your answer as to why Federation ships don't have cloak. If you want to play with a cloaking device, go make a Klingon.
I doubt that the Treaty made an exception for "ships with cloaks from alternative timelines." The last part of the argument here seems a little silly, although kudos to details of the first part. It's like saying that a captain from the mirror universe can come over to our timeline and start randomly killing our allies. His being from the MU would explain his motives, but his actions would violate the rules/laws/treaties of this timeline.
If the Klingons got Genesis in another timeline and came over to ours, would the federation allow them to use it as a weapon? There would be war.
So, the Feds are clearly violating the Treaty by using a cloak on a ship. It doesn't matter where they got the cloak.