I have fond memories of going to Chef Mickey's as a child and wanted to eat there as a bit of nostalgia, however my daughter is afraid of head characters when they get up close. Can you tell the hosts that you would rather the characters not come right to the table? And is it even worth going to CM's if the characters don't come up to you? I was thinking the monorail ride might be fun in addition to the lunch.
The handlers/characters are probably VERY used to all types of these situations, from my experience of what I saw. I wouldn't even worry about telling them beforehand unless you fear a full blown tantrum type of thing. They are usually pretty good at assessing the comfort level. They usually offer a wave and take it from there. Some kids eyes light up and they scream.."Mickey!" with a great big smile, and then the character knows it's "ok." OR, if they see the kid looking like they are trembling in fear, they might make a gesture akin to, "can I come over?" or something like that. I'm sure they go through training for situations like this. Have a talk with the daughter if you think it's appropriate (you might think it's a lost cause, I just don't know.) What I told my kid is that they want to come over and meet you, and make sure you are having a great time. They also would really like to take a quick picture with you, so you can remember they are your friends." I get it, some silent, goofy, giant headed thing with a plastered on smile can be weird. Totally weird. For the price of Chef Mickey's though, I'd be pushing to get the #$%^& photos. It's a major part of the cost. Good Luck!
You could still do the monorail ride, and go to one of the other places in the Contemp!
Have her sit on the benches or the opposite side of the table to the aisles. That way they won't be too close, and you can let her dictate whether or not she wants to get up and come round to high 5 them or even hug them. She can be as brave as she feels like being.
I've seen that happen and 9 times out of 10 the kids eventually come round. Especially if they see mom or dad having a photo with the characters first. Have an autograph book ready and at least make sure you get an autograph and picture of each character. When they come round again, hopefully she'll be up for it!