Whilst not really a character meal, Rendez Vous Des Stars at the studios has a chef who walks around with a trolley and it has a little Remy on a platter that talks. I don't believe there is a set time at lunch that he comes out but I did ask a waitress whilst we were there and she said that he does come out most days.
Lucky Nugget has a birthday tea party and a regular tea party, they are both on at the same time, so when you book you book which ever one you want, birthday tea party guests are seated in a different area to the regular character tea party, but everybody can see each other, just in different parts of the restaurant. We did the regular tea party and the birthday tea party at Lucky Nugget and the birthday one was much better, the characters came to our area first. At the birthday tea party, they give you a birthday card to get the characters to sign, a birthday cake for your party and the birthday guest also gets a talking mickey toy. (oh, the reason we prebooked and prepaid for both regular tea party and birthday tea party, was because when we first booked the regular tea party was at a different venue, so we wanted to do both venues with different characters, by the time we went to DLRP, they had combined the tea parties to the one venue, so as we had prepaid we went twice)
You can also order a birthday cake at the other restaurants I believe, we saw many of them at Cafe Mickeys and it was fantastic. This loud music would blast around the restaurant and then the cake would come out and all of the staff and characters would go over to that table and sing happy birthday. It looked really fun.