I'd personally look into a rental car, but it's probably not essential. I would also say that, with a 4yo, you might want to abandon the idea of going open to close. For us, traveling with 5yo, he was good from RD to maybe 2 pm, and then he was out. Tired, cranky, just done. Sometimes, we could revive him with some hotel time and maybe pool time, but sometimes we were just done with parks by early afternoon. We saw only 1 night time show on our first trip, and that was the night that we arrived, we got to MK around 5 pm.
In MK: DS has napped inside the hall of presidents foyer, it's very icy cool in there. If in strollers, there are quiet shady areas that might be nappable on Tom Sawyer Island, but that might be warmer. Frankly, Tiki Room is usually fairly empty and if the noise doesn't disturb them, it could be a decent nap.
EP: There's a baby station off to the left as you approach the bridge from future world to world showcase that would be a good area to nap. In between the voices of liberty shows, it's very cool inside the American adventure. It's fairly cool and dark just inside the temple in Mexico, though not necessarily quiet. If you're okay being outside in the stroller, there are usually benches or wall seating areas around the lake, especially near the transition from Germany to China (I think it's called the Outpost area).
AK: on discovery island, there are some "caves" that are dark and quiet and relatively cool. AK is a hot park and there tends to always be music playing, so it can be hard to find someplace quiet.