OK, this is going to be totaly my opinion. I think you will have to really see what each resort offers, and what you will really be doing at each resort.
I recently stayed at the beach club this past May. I thought it was very nice, but I will being staying at the Polynesian from now on.
I liked the theme of the resort, and it was very colorful.
I think the BC seemed much larger than the Polynesian. Our room was on the 4thy floor, very near the smoking area, and you could smell the smoke in our room.
The room was very nice, and very clean otherwise. The balcony was tiny, and covered in bird ummm, stuff... Thats ok, we didnt have time to sit on the balcony.
The Poynesain Balconies are much larger, and have always been free of bird items...
The restaurants in the BC were nice, but they do not have a fast food, quick item place to grab something to eat. They have the sit-down place, and in the morning, they have a small area which will serve you fast items, like bagels, juices, and coffees, and some cereals and fruits. They also have Beachs N Cream which was very cool. But unbelievably small.
The wait to get in the place was crazy...
We were able to get in , because we chose to sit at the bar area. Otherwise, the wait was over 45 minutes everytime we tried.
The Pools.... Here's the big one... For kids, this pool is great. It really not a pool, its a bunch of water paths that go around in circles, and it has some sand on the bottom. For kids, I think this is great. BUT, you have to share it with the guests from the Yacht Club, which makes it packed. It was always busy, and tough to find a chair. They do have a fast food place out near the pool, but it was only open until i think 6pm. (Might change for the seasons)
The quiet pool was closed while I was there, so I cannot comment on that.
Now, I am not saying that the pool at the Polynesian is not packed, it usually is during the day. But if you just want to grab a quick dip, they do have the quiet pool, which is usually kind of quiet.
I think if you went to Tikiman's page, you might get a better feel for what the Polyneisan can offer you.
Tikiman's Polynesian Site
As far as themeing goes... I think (my opinion) is that the Polynesian has better theming. You actually feel like you are away from the world here... Its different. I think the BC just felt like a hotel. It's a big square building.
For the Polyneisan, the lagoon is right there, and you can see the castle from the beach.
For the BC, it is close to Epcot, and in some rooms, you can see the fireworks.
I have been to the Polynesain with 3 nephews, but younger than yours. 5, 3, and 1... The 5 year old remembers everything about the resort, and still coments on the pool, the character breakfast at Ohana's, and taking the boat to the Magic Kingdom.
I think no matter where you stay, it wil be great for them, after all, it is the most magical place on earth!!
Have a great Time!!!!