How do I assert that I'm the right person for the role despite the family name?

Say it directly, without explanation. Kris Kluver, in The Dysfunctional Family Office, has Gail Mitchell tell a skeptical client. She told the client her last name and her gender weren't why she was running the business. She was running it because she was the best person to grow it. If that doesn't work for you, then you should walk. The line works because it doesn't apologize, doesn't justify, and doesn't ask the client to evaluate her credentials. It states the position and lets the client decide. Most rising generation leaders default to defending themselves with credentials. The credentials aren't the issue. The issue is whether you stand for something. Stating the position cleanly, without justification, demonstrates the leadership presence the client was actually testing for.