Lunch near the New York Stock Exchange
I may be wrong, but I don't think I could get out of Bobby Van's for $50 per person, or for that matter Harry's, although at Harry's I think I'd have a better chance. Check the prices before you decide. There is a place on Hanover Square (3 or 5, I think) with decent Italian. It has a first-name for it's name, something like Joseph's. And it also might fit the bill.
Cannot comment on Brouwers, etc.
Bobby Van's is right across from the Stock Exchange, on Broad Street I think. Perhaps their more casual lunch menu would fit your budget.
On Stone Street (a 5 minute walk) there are a few options. I have eaten at Harry's several times since it was renovated, and enjoyed the experience. Also, Brouwers of Stone Street was was recently noted in the Times, though I have not yet eaten there. There are several more casual options (Ulysees, Adrienne's, Smorgas Chef) as well, though reserving for 8 may be difficult. Good luck.
I can think of a number of places you might fall within that budget, but would be at risk of not. I could also think of someplace like Michael's on Broadway, where you would probably never go over, and which might work. But my recommendation would be to cross Broadway and walk to Greenwich St., then eat at Caracello's, which is in the block north of Rector. Since, 9/11, it is never crowded and it serves good but traditional Italian, and it is accomodating. When a client wanted sole cooked Livornese, which was not on the menu, it was not an issue at all. It's not the most upscale place, but it should work and fit your budget.