Giangilberto from Milan here. I can't referee the two hotels for you — I haven't slept in either, and I'd be inventing if I pretended otherwise. But I know both spots well, and here's the thing: you're not really choosing between two hotels. With one evening in Naples, you're choosing between two completely different anniversary nights, because those addresses sit in two different worlds.
Piazza Bellini is the living room of the historic center: cafés and aperitivo tables on a little square with ancient Greek walls in the middle, bookshops, conservatory students, and the whole Spaccanapoli maze starting a few steps away. Evening there means noise in the good sense — you're inside the city. The honest flip side: "inside the city" lasts all night in Naples, so ask for a courtyard-facing room, and the arrival by taxi through those alleys is its own adventure.
Galleria Umberto is the opposite scene: the grand nineteenth-century glass arcade by the Royal Palace and the San Carlo opera house, elegant, orderly, catches its breath at night. From there you stroll Via Toledo, or have a coffee at Gambrinus like the Neapolitan bourgeoisie has for a century and a half. It's the polished, romantic Naples — some find it almost too quiet after dinner.
If it were my anniversary and my one night, I'd honestly pick the Bellini area — Naples' magic is in its chaos, and with a single evening I'd want to be swimming in it, not looking at it from the balcony. But if your idea of a celebration is a beautiful dinner and a calm elegant walk, the Galleria is the safer romance.
One logistics note that matters for your plan: for the early train to Rome, both are fine — ten to fifteen minutes by taxi to Centrale, just book the cab through the hotel the night before rather than flagging one. And for the anniversary dinner, wherever you sleep: reserve, even in Naples, even on a weekday. Congratulations on the five years!