Thanks to all of you for the great information. Thank you especially, Birdsall, for the compelling synopsis of "Traviata." I'm quite excited to see it now!
The issue of where to see it may be resolving itself. I realized that, while I am in St. Petersburg, the Mariinsky will be performing it in the new theater. (I mis-read the schedule before.) However, when I am in Moscow, the Bolshoi will be performing "Traviata" in their historic theater. Since I was also having problems figuring out how to see a performance at the historic Bolshoi theater, it seems like maybe "Traviata" in Moscow is the right option. (Does anyone have an opinion on the Bolshoi's opera company?)
Separately, I'm still struggling with whether or not I should go to "extreme measures" to see the Mariinsky while I am in Moscow. I'm actually thinking about taking the fast train back to St. Petersburg, seeing the Mariinsky's season opener ("La Sylphide") in the historic theater, then returning to Moscow on the night train. Of course, that would be expensive and possibly exhausting, and to be honest, "Sylphide" is not my favorite work (with the disclaimer being that I have only seen it once, and it's possible I just didn't like what that company did with it -- I'm guessing it would be better at the Mariinsky). The Mariinsky will also be doing the "The Little Humpbacked Horse" during that same week, so, as an alternative, I could make an extreme side trip to see that performance, thought it would be in the new theater. But at least I would get to see the company perform before leaving Russia.
I have to admit, it has also crossed my mind to push the whole trip forward two weeks in time so I can see more of the Mariinsky. But I had separate reasons for wanting to be in St. Petersburg in September, and I'm also now kind of curious to see the Mikhailovsky performances slated for when I'm there. And I notice it's easier to get front-row seats at the Mikhailovksy.
Thanks again, everyone, for the helpful input. I love this forum!