Personally I would let your DH adopt him. Have you asked your son what he wants? Is he aware that your DH is not his biological father? Changing his last name shouldn't affect him learning about that part of his heritage and where he came from. esp if there are other family members to help. and he still is keeping his middle name right?
Although we're not biracial, my sister and I have different fathers. When I was born, my sister was 5 and soon after I came my parents got married. My mom explained to my sister that she would be taking my dad's last name and that if Rachel wanted to, she can take his name as well. It was a big step. My dad loves her and takes care of her and her bio dad wasn't in the picture what so ever. My mom said when my sister came up to her later on that day and said yes I would like to have the same last name, it was like Rachel had accepted my dad as her dad. Now as she got older she did ask about her bio dad and my mom was willing to give her the last information that she knew of him. In the end it was a dead end since it had been so long and even then, if she found him, he wouldn't have replaced the true dad who helped raise her.
So yes, if he wants to, I would let him change his name, and teach him about his culture too. But in the end a Daddy isn't who's sperm got to the egg, it's who makes you eggs in the morning because he loves you