When DH suggested we split in '97 in order to try WL, I was very much against it. We had a small child that needed naps and it just seemed like a pain. It ended up being so painless and so much fun that we now split every stay.
Some times we arrange it for a non-park day, other times we pack up and change hotels early. Check-in and head off to the parks in time for opening. If the room isn't ready, we check back with the resort until it is ready. We now have 2 little ones who both needed naps and have never run into a problem. We always have a rental car, so we move ourselves.
One word of warning, Disney won't guarantee your bags will be in the new room until 3pm (I think). Therefore, if you aren't moving yourself, I would take a small bag with anything you might need during the afternoon. That being said, rarely have I heard that someones stuff wasn't there when they needed it.
Do it, it's a lot of fun. It makes it feel like two different vacations.