To try out the "rest" time, I would say bring him to the mall one day, and then try to get him to lay his head down on your lap and rest on one of the benches. Some kids will be flat out and have no problem shutting the world out and chilling. Some kids can't stop listening/watching/being engaged in their world enough to chill.
Probably not a popular answer, and I'm by no means an elitist, but, this is why I paid all that money for the monorail hotels. I wanted to get to and from the MK+Epcot easily for naps. Even at age 7, my pair took full on naps in the afternoon if we were going to be out late. Shoot, I took one myself. We didn't nap every day, but any day we had evening plans, we napped.
Since these resorts can be cost prohibitive to many, my advice is to just go to your resort for a nap. It might cost you 3-4 hours, but, trying to get them to chill out and actually rest in a public space might cost you nearly that anyway, plus aggravation on both of your parts.
Some kids can go all day long, and never need a nap/break/rest. My kids are not that way, some are, some aren't. Since you already realize your child will benefit from a rest, I would spend the time, considering it an investment against meltdowns.