I e-mailed the U.S. Consulate in St. Petersburg with a question about passport renewal, and they responded very promptly.
This is what they said: "Thank you for your e-mail. You are welcome to apply for a passport renewal. Since your current passport is valid for less than two years, we can return it to you without canceling."
If that's true, then it's possible that I could get a new passport, then keep the old passport with the 90-day visa, while applying for a new LOI with the new passport. That would be very convenient.
I'm hoping to get this taken care of as soon as I get back to Russia, so that I'll have the new passport for starting the next visa process.