It's been my experience that in K the teacher or aide takes all the children to the bathroom on a regular schedule. Is he not going then? I'd talk with the teacher to find out what happens in her room. If he's not taking advantage of these times, ask the teacher how she can handle it in her classroom. Does she insist that he go? I'm sure he's not the first one to have difficulty.
Instead of giving him a consequence for having an accident give him a consequence for not going to the bathroom when you tell him to do so. He should not have an option of going to the bathroom or not. He must go and there will be a consequence if he doesn't at least try to pee or have a bowel movement.
My grandson had a lot of difficulty with constipation at that age. He also took Miralax. I suggest that Miralax will not give him a runny movement. I wonder if he doesn't have a medical issue such as irritable bowel syndrome. Or perhaps he's allergic to a food or has a lactose intolerance. Ask the doctor about that.
Keep a record of what he's eaten and of his activity prior to having a loose bowel movement and see if there's a pattern. Note every bowel movement, and its consistency. Food and emotions can affect movements. See if there is a pattern.
I would not make staying dry a way to earn anything that he already has the ability to do. Perhaps reward him for staying dry by giving him something extra. Keep a chart and after a week of being dry take him for an ice cream, for example.
Talk with him about ways in which he can help himself stay dry. Talk about how his body works. Tell him you're going to help him build a routine so that he can stay dry.
As for bowel movements, have you tried having him sit on the toilet 30 minutes after his evening meal. Or, if you have time, after breakfast. Make this a daily routine. Take him by the hand and lead him to the bathroom. Make it fun. Perhaps have toys in the bathroom with which he can only play while sitting on the toilet. Read to him. Tell stories. Start a story, have him add something, you add something, he adds something. Sing.