Jerome Bixby did pitch a "Mirror, Mirror" sequel to TNG called "Broken Mirror," but it was rejected. I'm sure they got a variety of "evil twin" episode pitches, since they heard hundreds of pitches a week and no doubt got every cliche and TOS rehash imaginable tossed their way multiple times. But it isn't the concept that makes a story worth doing, it's the execution. Presumably nobody found an angle on the evil-twins idea that made it worth doing.
Besides, for most of its run, TNG was trying to establish its own identity as a distinct entity from TOS. Aside from doing "The Naked Now" as episode 2, it mostly tried to avoid reusing TOS elements and to develop its own story threads and species. That changed over time, as TOS fans like Ron Moore came onto the staff and began injecting more TOS elements, but there would've been resistance to doing a Mirror Universe sequel just for the sake of doing a sequel. Presumably what made the DS9 producers think "Crossover" was worth doing was that it deconstructed Kirk's solution in the original episode, showing that it may have done more harm than good. That was the kind of fresh twist that made it worthwhile in a way that a more-of-the-same Terran Empire revisit would not have been.
Perhaps B&B were too early in their stint to run so low on ideas as to create an evil crew show or arc.
I will never understand the "B&B" myth, the idea that Berman and Braga were perpetual collaborators. Brannon Braga didn't even join the TNG writing staff until season 4, when he was a staff writer and intern, the lowest rung on the totem pole. He made it to story editor in season 6 and co-producer in season 7, the lowest rank that actually has "producer" in the name. And as a writer, his usual collaborator on TNG was Ron Moore. (The actual showrunner on TNG was Gene Roddenberry in season 1, Maurice Hurley in S2, Michael Piller in S3-6, and Jeri Taylor in S7.)
Braga then moved to
Voyager as producer (one step up) under showrunners Piller and Taylor, moving up through the ranks to supervising producer in season 2 and co-executive producer in season 4 before eventually taking over as executive producer/showrunner in season 5 and 6, then ceding the showrunner post to Ken Biller for season 7 while he moved on to develop
Enterprise. On VGR, his main writing partner was Joe Menosky.
As for Rick Berman, he was more the production executive than the showrunner (i.e. more the equivalent of TOS's Herb Solow than Gene Roddenberry, for all that he was treated as Roddenberry's successor), and he rarely contributed as a writer on TNG, DS9, or VGR, except for developing story outlines for a lot of the big 2-parters. The only actual scripts he wrote on any of those series were the TNG episodes "Brothers" and "A Matter of Time," both of which he wrote solo. He worked closely with Braga on the two VGR seasons that Braga ran, but as Braga's boss, not his partner. He co-wrote the stories to
Generations and
First Contact with Moore and Braga, but worked on
Insurrection and
Nemesis with zero involvement from Braga.
The only time when "Berman and Braga" existed as a real collaboration was on
Enterprise, where Berman moved into screenwriting on a regular basis and he and Braga served as co-showrunners for the first three seasons. Since ENT ended nearly a decade ago, they've gone their separate ways and haven't worked together since.
So if we're talking about TNG, for the most part, you should be talking about "B&P" -- Berman and Piller. At that point, Braga was basically the Wesley Crusher to their Picard and Riker. Or maybe the Ensign Ro, considering that he only came onboard about halfway through the series.
