III Forks has nice private rooms ~ we have rented them out for a graduation & a birthday. My husband has his work holiday party there almost every year. Maggianos is another place that has private rooms as does Patrizios.
Not a private room - but it's my "go to" place lately for large group functions lately. Fede Trattoria on Eldorado. They'll put your group in the back corner on a long table. Sort of a separate wing of the restaurant. (NOLA Grill does this also for groups btw.) Plenty quiet and private for a business meeting - particularly on a slower week night.
We hosted my son's graduation party there for around 25-30 people and it was plenty private for us - and the bill was reasonable even though we served a full dinner and alcoholic drinks. Service and food were phenomenal.
ETA - oops I just read the OP again - both Fede and NOLA are closed on Mondays. Never mind.