This past April we had a split stay: AKL the first weekend, then 6 more nights at BWV.
The actual move is pretty easy. You pack up your stuff, call bell services and tell them you are changing resorts upon check-out and need the baggage transferred to another Disney resort; they pick it up (tipping appreciated), label it, load it on the luggage trolley and give you a receipt. The you go off to the parks for the morning. Early afternoon (2-3) you head to your new resort, check-in, then visit bell services there to see if you luggage has arrived. If it has and you room is ready, they will meet you in your new room and unload (again a tip, these are different bellhops after all). If your room is not yet ready, then they will hold the luggage and deliver it after you get into the new room. If it has not arrived yet, you leave your room number with them and they will call you before they bring it up.
Now the PITA part is that you become essentially homeless for the middle of the day, without access to your stuff. If your kids will nap (always a good idea in the excitement of WDW!) you will not have a room to go back to until your new room is ready. We always take an afternoon break (sometimes a nap, sometimes a swim) in the afternoons at WDW, when the parks are hottest and busiest; so a split stay makes that impossible unless by some miracle your new room become ready early.
The other PITA is just having to pack up your stuff from the first room, and not being able to really settle in.
But then, a split stay can be a really nice way to get a deluxe stay in on a budget. Perhaps you could do a Value or a Mod before your AKL stay. And I emphasize the cheaper room before your deluxe stay; you want to finish off the vacation in style, don't you? The other way, starting at AKL, you will be missing it when you downgrade to the Mod or Value.
Good luck with your planning and decision!
BTW, I am a HUGE AKL fan. It's not for everyone, but our family adores this resrt and everything about it. When we check out, our one regret is not having spent more time just hanging around the resort. I hope you really enjoy it!