We've stayed at both with small napping children. We're not super big fans of BCV but are HUGE on AKV right now for our kids ages 19 months & 4.
We were at BCV 1 bdrm for 9 nights our home resort is BWV (which has the exact same set up for rooms we've been members for 10 years now) BUT before

we had kids I didn't appreciate crazy door configuration of the 1+ bedrooms. We have a double stroller and we could barely get the thing in the door there's so freaking much furniture in the way and then it felt like we were all over each other. It seemed like I ran into the dining table with my hip 15 times a day. We were always running after DS#1 making sure he wasn't slamming doors (3 doors to get to the potty, bang, bang, BANG). When the baby was sleeping in the bedroom portion you cannot get into the bathroom with out leting light in from the living area. The pool is great but busy I never really feel relaxed. We weren't super excited either about the sandwhich counter set up it was always crazy busy in there and there's no where to sit and eat, so you buy food, haul food. The premium
point charts don't really endear me either since the views can be less than fabulous, at least at BWV you get to see a pool or BW, not premium points for parking lot. The walking to Epcot and DHS is nice, for the other parks we would valet the rental car so one less thing deal with coming in and out of the resort.
We stayed in a 1 bdrm the 2nd week were there the same trip at AKLV and fell in love and are returning for our next trip. It was nice to be able to compare 'living' in each right back to back. We love that the bedroom can be closed off and you don't have to go thru to the bathroom and the potty is better arranged in the bathroom. The closet in the bathroom is awesome for storing toiletries away from little hands vs the low cabinets in the rest of the DVC resorts, they won me on that! We felt like we could make all the niose we wanted in the living areas the baby never heard a peep. We were glad we took a monitor so we could hear him! When the shades are pulled the rooms are super dark. When we took the stroller with us it was a straight shot into the room and there was a nook where we could fold it up out of the way. We also liked having the actual food court. The pool to us has a much more relaxing vibe to it. Altho it is a ways away we always rent a car using the valet service and never take the buses so it doesn't make the difference to us and can't comment on how timley it runs. Seems like there have always been many there.
There are pros and cons to both resorts. Things we adore about some resorts pre-kiddos have become pains with young children. My parents try to stay at BCV every trip they LOVE it and we always stayed at BWV & WLV as our family grew our needs changed.