Although there are beach resorts in Cocoa Beach near Cape Canaveral, nothing comes close to the style and comfort of the Four Seasons Orlando (if you've read Tom Wolfe's The Right Stuff, it hasn't much changed), so your best bet is to drive out and back the same day. Your hotel concierge can hire a driver if you prefer not to rent a car, but if you're linking Disney with some time in Palm Beach or Miami, book your rental for pick-up in Orlando, drive out to KSC then head south afterwards.
First things first: buy ahead! You're not the first parent who wants to play astronaut with their kid, and tours do sell out. Everything can be easily purchased online, though, if you can wade through the options. Many great exhibits are included with your visit, but adding on the VIP tours can get you into normally restricted areas. Skip the Then and Now unless your child is older and can handle a three-hour tour, and go for the Up Close option, which puts you on a tour bus (ie: not lots of walking) and takes you to the launch pad. Selfie time!
There's also a family Astronaut Training Experience program, which runs every other Sunday and is open to children ages 7 and up. Lunch with an astronaut gets more mixed reviews. You'll be part of a big group, so consider carefully if it's really a priority.