I think this is about what will make you happiest in the long run, not what you "should" do. You don't owe him anything, and if you truly do not want to communicate with him at all, you don't have to. But if selling his board without letting him know will make you feel like a heel, that's not a great situation either. You may feel an extra need to be decent, since you broke it up.

You could write to him just to let him know that you're selling it, but repeat that you are not going to engage in a dialogue about it. Communication is two-way, after all. If you think that will be impossible, I think it would be better to not let him know. Sometime you just have to live with the knowledge that somebody thinks you're mean.

Another option could be to have a friend, maybe the friend who has it, do both the selling and informing your ex about it. That would remove a hope he might have of engaging you in conversation. Depends if you have a friend who owes you a favour