With little ones, we specifically look for a zero entry pool area and/or a splash pad, a decent playground (covered or shady, if in summer), an indoor activity room, maybe mini-golf and/or theater room. Check the TUG Reviews to see if the features most important to you are available.
In II, look at Marriott Harbour Lake, Marriott Grand Vista, and Marriott Lakeshore Reserve, for example. Sorry, I'm less familiar with II options.
In RCI, look at Wyndham Bonnet Creek, Silver Lake-Silver Points, Vacation Village at Parkway, Summer Bay, and maybe Mystic Dunes HVC (watch out for the added resort fees). Also, since you're only planning one day at the parks, look at Wyndham/Worldmark Reunion and HIVC at Orange Lake-River Island, since the drive to the parks is further but they offer a ton of kid-friendly activities.