In my experience, I know
Diwan (Diwan al Sultan Ibrahim) has a good quality of food. It's just a regular Lebanese restaurant. The place is big, and they have been around for a while. They have a branch in Geitawi or not far and one in Antelias. I was once impressed by the foul dish, but it was a long time ago. Anyway I was never disappointed by the quality of food there.
Not exactly cheap but you may be able to manage within your budget.
If you want cheaper and authentic you can try le Chef in Gemmayze (much smaller). I think you can even divide your budget by 2 there. It is popular amongst tourists.
I haven't been at either for a while. I'm not fully sure about the new prices, but these are old restaurants and I bet that the quality is reliable and that they still roughly follow the same formula.