I've never tried this, but I think this should be possible. First, you would need to open up
MDE in two different browsers (not just two different tabs in the same browser). Then, in the first browser, select the option to modify an existing fastpass and select the people (person #1 and child under 7) and fastpass you are changing. Search for A&E and see if a better time shows up. In the second browser, do the same for person #3 and see if you can find an overlapping time. If you can't find anything, just click cancel on both, and nothing should change. This way you can see if there are any overlapping times without having to commit to any changes before you are sure the times you need are available.
It may be possible to search for 3 single person FPs this way as well because I think I read somewhere that once you make a child's initial FP, you can then change it w/o being connected to an adult, but I'm not sure about that.
Again, I don't have any real experience with this, so feel free to correct me if I am wrong or if there is an easier way to do this!