I love the AKL. Howevre in your situation it might not be the best bet. My guess is that you'll spend the most time at the Magic Kingdom, so being closer to that might be a better bet.
The Polynesian has some buildings that don't have balconies on the second floor, although if you make the request for a ground floor room due to children when you book, they're good about granting that type of thing.
The lush grounds have plenty of foom for your older child to run off steam, a nice beach, and a great zero entry pool.
It's a quick monorail ride to the MK, and unless it's really crowded they won't make you fold the stroller.
The cast there are wonderful, and the resort offers a variety of amenities and some nice shops. I believe there are now mini-fridges in every room.
Regretfully none of the moderates have a balcony. They all have exterior entrances, so the "blacony" is the walkway outside your door. Not exactly a quiet place where you can leave the door open to hear while baby naps.
If the Polynesian is out of your budget, keep the reservation at the AKL, it's incredible--the resort envelopes you like a warm blanket and you won't want to leave. Be sure to book a savanah view room, watching the animals is very peaceful.
Anne