I think you are overthinking this, H.. If you were considering full-day at a private school because you like the idea of full day K, then you aren't running away from a problem, but going toward something you feel would be more beneficial in the long run.
I'm not sure what you want to be done, given the situation? If it were me and I believed that the shorter day program was the better option, then I would discuss the previous situation with the prospective K teacher. I do say: I have to wonder what steps the current daycare staff is taking to curb the bullying/upsetting behavior while the boys are there, besides writing them up. When we had a kid who was having problems being kind/getting along, we'd really keep a close eye out for mischief and try to get in there before it happened as well as making an especial effort to keep those kids within sightlines/earshot at all times. It won't be 100%, but when you know you have a child in the group who is a negative influence, that vigilance really helps.
Keep focusing on correcting your son's behavior whenever necessary. You can tell him "We know (kid's name) is not safe or kind with their body/words, and if he tells you to do something, check with the teacher to see if it's okay first." If it were me, I'd want to know what the discipline piece was at daycare (what are they doing in the moment to deal with this behavior within the group). Really, though, if you are already thinking full-day would be good for your son, go ahead and enroll him in the private school. Don't let your sense of 'oh, we've got to fix this first' stymie what you already felt was a better option for your family.