The closest thing to the real thing is the
Vatche U113. And it's literally like...Tiffany will need to see the stamps to know. There are people who will tell you that the "only upgrade" thing is not really true--I can think of at least one poster on here that's upgraded twice.
You can get the U113 from HPD, WF and GOG. BGD does not work with Vatche (Full disclosure: I personally am not a HUGE fan of theirs myself, having owned a different one of their rings prior to this and it was NOT a good experience in the workmanship department. I am also not Whiteflash's biggest fan either from a prior really bad experience, but other people really love them, Christiflora among them.)
But have you seen my ring? It's pretty bang on. I have a narrower shank (I have the
BGD tapered setting and they have two
other versions) than the original but if you didn't know that, you probably would never know it wasn't from Tiffany. Many of their SAs can't even tell when I go in there with my real Tiffany wedding band next to it.
RE: your other point, you likely can do pretty well going non-branded, meeting all your specs, and still keep your original. Though going non-branded gives you a world of options for the setting as well--by that I mean, you could have the stone set higher if you wanted to, or get a halo even (you want finger coverage? That's the way.)