I've stayed at the Beach Club many times and I like it because you can easily walk to the World Showcase entrance of EPCOT, and you can also walk to Hollywood Studios (formerly MGM). The pool (Stormalong Bay) is fabulous, the hotel is beautiful, and there are buses that go regularly to Magic Kingdom, Animal Kingdom and the water parks. You can also hop on a boat to get to EPCOT or MGM if you don't want to walk.
My next trip, we will stay at the Poly because I stayed at Shades of Green and it's a short walk to the Poly, so I got to see a lot of it and I really liked it because it reminds me of when we lived in Hawaii (and the Beach Club reminds me of the east coast beach resorts, where I grew up). You can get on the monorail at the Poly itself, and it will take you to the other resorts in that area (Grand Floridian and Contemporary) and also to Magic Kingdom and the TTC, where you can switch lines and take a monorail to EPCOT. You can also take a ferry to Magic Kingdom, which is fun (I think anyway -- I'm just a kid at heart -- LOL!).
I think either hotel would be a fine choice, and you just have to ask yourself which one will be best for your specific family. Our stays at the Beach Club occurred when my kids were 10 and up and they loved Stormalong Bay, and we usually went during Star Wars weekends for my son, the Star Wars fanatic. My other son, the daredevil, wanted to go on Rock-n-Rollercoaster and Tower of Terror. I love many attractions at the studios (I even got on "Who Wants to be a Millionaire -- Play it!" once!). If you think your kids, because they are so young, will want to spend the most time at Magic Kingdom, the Poly might make more sense because of the ease of getting there (ferry or monorail). You could easily pop back to the Poly for an afternoon nap or rest and not lose as much time as you would getting from MK to the Beach Club. But if you think your kids would enjoy spending more time at EPCOT or the studios, then the Beach Club would give you that convenience. So, it really depends on where you think you'll spend the most time.
Either hotel, though, will be wonderful and I hope that you and your family all have a magical time!
Oh, and yes, you can easily get from the BC to Extra Magic Morning hours at any park, and if you walk to EPCOT, though, you will go in the back gate and have to walk to future world because only future world is open for morning EMH. But you can do it -- we did it lots of times. And, of course, the boats and buses run before EMH, so you can even make EMH at Animal Kingdom, Magic Kingdom and MGM without any problems.
-Dorothy (LadyZolt)