J.,
I'm a former CM from the Toontown "tents" years ago. I hesitate posting this lest hoardes of people will be begging for private meets and saying they deserve it because they heard so-and-so got one. So, people... chill. (only speaking from experience... you make one announcement that 'infirm?' kids don't need to wait in line, and all the sudden every kid is claiming to be on death's door)
This wasn't something we were ever told to do, but every once in a while, we would backdoor people for private time with Mickey. It would only be a few minutes, but much more relaxed than the general line. And you got the whole room to yourself. Generally, you go in with a group of people, and only the last of each group is actually in the room alone with Mickey and his aide. This is back when the lines to see Mickey were 1-2 hrs in the middle of the day; don't know what they're like now.
Anyway, we would do this basically for VERY special occasions, such as Give Kids The World, engagements (as in the actual "Will You Marry Me?", not just for engagement/wedding parties), and super-psycho Mickey fans who were in tears because they waited all their lives to meet him and they spent hours at home making special gifts for him.
No charge. Sometimes the guests asked (engagements), and sometimes we would offer. Some CMs couldn't care less and wouldn't do it, regardless.
Honestly, it sounds like you can get equal interaction with him at the DD photo shoot, plus it's guaranteed, and you'll get a professional photo, rather than relying on a CM who may or may not know how to work a camera.