Awww, that's simply awful!
If you can, I would personally try to go to the first show possible on Friday (or possibly the second). I'm not sure that crowds are generally lower for the first shows (I think they might be worse in some cases). But I wouldn't be surprised if locals/daytrippers/tourists taking long weekends came down to see the show sometime well after park opening on Friday. A local who took half a day off work would be in great shape to see the 4:15 or 5:50 shows.
I know right now people are reporting being able to get in while getting there shortly before showtime (we got there practically AT showtime on an overcast day in December and managed to get in the lowest level). I would keep a close eye on the DISboards this summer and see if that is still the case as the final date for Aladdin approaches.
Between blockouts on Saturday and human nature, my best guess is that the very worst crowds would be the last day it runs (particularly the very last show). But I agree with you that it should be pretty busy the whole time you are there.
At this moment my guess would be that you'd be fine on Friday if you got to an early show an hour in advance. It will be fun to see how far off I am in August.
Have a great trip!