Hi Rich - so…
La Pergola - https://romecavalieri.com/la-pergola/ winning no prizes for ‘cheap’, it’s a 3*, with cost and experience to match. Wine still better priced than in the UK though, I’d say (Italian, anyway).
Per Me - https://www.giulioterrinoni.it/eng/the-restaurant/ - it’s a 1* with excellent food. Wine list a healthy selection, and some (though not all) the ‘names’ are marked up sensibly - e.g. the Tig is under 1.5x UK shelf-price
Babette, on Via Marguta - I’m quite fond of this place. The food is what it is - well-done home cooking, but no awards to be won - but crucially a seriously good wine can be yours at a sensible price; worth a visit.
Ristorante Ad Hoc, on Via Ripetta - good food, good wine, nothing not to like. Extensive wine selection; not as keenly priced as Babette, but wider selection, still decently priced, and the food & wine together are a different story here.
I think I recall Il Gabriello (Via Vittoria) being decent enough, but it is harder to tell from online photos if it’s the right place than I thought it would be!
As ever, viamichelin is a great place to start searching…
You’ll find good wine bars around too, terrific selections by-the-glass, tempting bottles to bring home, good small plates to nibble - nice way to have a gentle lunch or starter before dinner.
There are many more great restaurants that I can’t remember the name of, and many, many more I’ve never visited - have a great trip, and do report back!