You might want to try a place like Auberge Communale de Satigny (it's better than it sounds). Since it's (just) outside Geneva it should be cheaper and by now the vineyards are getting green so the environment is quite nice.

Other options I can think of are: 

* No Pizza (pizza) - they should be able to manage a big group easily, though if you include starter and drinks you'll probably go over budget.

* Cafe Restaurant de l'Aviation (near Meyrin) - big enough to handle groups and quite cheap, but they basically only have a single menu choice (beef with a spicy sauce and fries). 

I have had groups of up to 24 at La Ruota restaurant on rue Industrielle in Acacias. The penne al limone is exceptional as is the service. The nearest tram stop is Industrielle on the 15 line and it's a scant couple of blocks away.
The menu will definitely fit your budget.

We just held an event at the Resturant des Nations, near the UN. They don't have a website, but you can call +4122 734 35 45.

The staff was amazing and catered to every request (having diabetic, lactose/gluten free, vegetarian, and nutless options). They offered us a 3 course menu for 45 CHF's a person, but I'm sure they would offer a cheaper option. 

Would you consider a pizzaria around servette? If so try La Primavera. Excellent pizza, and they have a side room reserved only for groups.

Au Petit Chalet near Cornavin is very good.