It wasn't an issue until Worf had to get all meta in Trials and Tribblations and explicitly acknowledge the change as something in-universe.
I think that an interesting way to have handled the difference during Trials And Tribblations in the appearance between Klingons in the 23rd century and Klingons in the 24th century would have been for in the beginning of the episode (in the 24th century) have Worf look the way he always does, but then when the Defiant crew is in the 23rd century Worf looks just like all the other 23rd century Klingons, and no one think this is unusual or comments on it.
Then when the Defiant return to the 24th century Worf is back to his standard appearance. Again no one on the show notices.
But then don't you think that fans would want to know why Worf magically transformed? Wouldn't fans want to know how our heroes somehow didn't notice such a radical transformation? While your idea is original, I think it would open up more questions. Remember the fanbase you're dealing with here.
It would've been simpler to have Worf not even go aboard the space station in the first place. I can understand that it probably to good of an opportunity for the writers to pass up. Enterprise maybe wouldn't have had to address the issue if DS9 didn't bring it up..