I like the pizza at Taverna Le Coppelle, but they actually do try to turn their tables quickly. It sounds as if you want something a little less frantic. I suggest looking into Armando Al Pantheon, by the Pantheon (as you might have guessed). It gets raves from people whose opinion I trust.
Here's a comment from TripAdvisor about them:
Despite its location near the Pantheon, Armando al Pantheon is not a tourist restaurant. Chef and owner Claudio Gargioli and his family gracefully serve quality cucina romana at moderate prices seasonality as their main concern. I reccommend typical Roman staples like pasta alla gricia, amatriciana, cacio e pepe, or the delightful pasta e ceci soup. In addition to meat entrees like roast lamb, saltimbocca and trippa stewed with tomato sauce, pecorino and mint, or the delectable pajata, Armando al Pantheon also cares about pleasing vegetarians, who can choose from their own separate menu. There’s also a good wine list.
And I would be very leery of trusting recommendations from your hotel, esp. if it's a hotel of any size. The staff at the front desk of many larger Italian hotels get kickbacks from restaurants if they send someone their way. This will be esp. true if they tell you to tell your waiter who recommended them.