If your child no longer needs a nap, pull them from the home daycare and wrestle it out with a center that is probably going to tell you your child needs to lay down for a certain amount of time.
Not every environment is right for every child. We do a ton of arts and crafts in my home. A couple of projects every day. It's what I do. I would not make and exception for a child that was into sports or basketball or soccer or whatever to go outside and play their choice of sports while the rest of us did the art activities. If that child or their parents were so opposed to doing what I had scheduled, and they knew that was the type of program I ran, then they need to find care elsewhere.
Nap time is a scheduled daily activity and it is usually written about in every home providers policy so the parents should know about it in advance. If they do not want their child participating in any of the scheduled daily activities then the child should not be in that program. The provider should not be and is not obligated to "make exceptions" for anyone.
Kid age out of programs all the time. To the woman who left and put her kid in a Montessori program - you probably did your provider a favor and saved her the time and energy of finding an excuse to terminate you. Home providers don't really have the luxury of worrying about the best interests of a particular child, it's the best interest of all of the children. If your kid is the square peg and going to screw up the nap routine of all of the rest of the kids, YOUR kid needs to go somewhere else. Not really an issue.

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