I wouldn't be so quick to LABEL a five-year-old as a bully. There's a difference between what he does and who he is. That doesn't mean his bullying behaviors don't need to be addressed. They do.
Aside from the discussions and talking with the teacher and other parents, I suggest one of you be present at recess, where you can observe him. (You can take turns.)
However, I think it's better to volunteer to be a playground parent, meaning organizing games, etc, where you are able to interact, coach, and verbally acknowledge him AND other children for playing well together and learning new skills.
I'd also make sure he's getting a lot of active play time after school (and before if you can manage it.) Boys, especially, need to run around and let off their energy. During Covid time, children have been penned inside, without the normal outlets to physically MOVE. In addition, they exposed to a great deal more stress than normal.
A good afterschool activity is martial arts. A reputable studio:
1. Heavily stresses NOT fighting or bullying. Supervised sparring only takes place at the studio.
2. Teaches children positive values and safety precautions along with important physical skills.
3. Provides a supervised outlet to discharge energy.
Before you sign him up at a studio, observe a few classes and speak with the instructors to learn the ethics of the school. Then speak about your specific concerns. The instructors will work with him on the bullying issues.
Debra Holland, M.S., Ph.D, M.F.T (and second degree black belt)