You could also try some of the Downtown Disney restaurants & those in the Swan or Dolphin. I agree that you most likely won't find anything for a group that size in the parks, & at most Disney hotels. If you go at an off time, you can probably get seated at T-Rex or Rainforest Cafe. Naturally, you'll have to sit at separate tables, but I'm sure you expect that. Both of these restaurants have a fun theme. Raglan Road may be possible at off times too. You could try calling all these restaurants directly & setting up a reservation. They don't book all their ADRs through Disney, so you may get lucky. Planet Hollywood & Wolfgang Puck Cafe may also be possible at off times.
For a character meal, you could try Garden Grove at Swan resort. For burgers & ice cream, try Fountain restaurant at Dolphin resort. Fountain is similar to Beaches & Cream. I'm not sure what their seating capacity is though. S/D have other restaurants you may be interested in also. You can easily access them from Epcot or DHS. I would try calling the S/D directly to make an ADR for that many people.
Good luck finding something. I know you'll have a great time regardless. I used to travel with sports teams, when my niece & nephew played. We had a lot of fun times.