When the first show ends, three things will begin to happen at the same time:
1 - people will start to leave the area
2 - people will be entering the area
3 - people will stay put to watch the fireworks
You actually want to arrive in the area before the first show ends. I suggest arriving mid-way through and hanging the New Orleans Square shops. When the show ends, 9:23pm, you come out and work your way towards where you want to sit while the above 3 things are also happening.
The fireworks will start at 9:25pm. Watch them as you are finding your spot. With the group you have described, I would suggest the seating area by the riverfront. Go dead center of the house on the island, in the roped off seating only area, back row. If you are in the back row, you will not have to collapse your stroller and you will be able to see fine as everyone must remain seated once the show starts.
This is not optimal viewing for fireworks (compared to Main Street), but it is the best way to view the fireworks, get a "good" spot for the 10:30pm show, and NOT deal with the flow of mass of people from Main Street to the riverfront.
Enjoy,
Dreams