Call to find out! Something may be open if you aren't too picky with days or times. But to get better odds, why don't you broaden your scope of restaurants. Ohana's is very good and it's about half the price of HDD. Whispering Canyon Cafe is another choice. I know they aren't dinner shows, but both have plenty of entertainment. I know that Chef Mickey's does get booked very quickly, but you can try something else like Cape May Cafe for breakfast or Liberty Tree Tavern for dinner. Of course, if you have your heart set on HDD and CM I'd give a call, and if you can't get anything, I'd call every day looking for a cancellation.
Good Luck!