Standing Your Ground Respectfully
Had a disagreement with my brother in law about how to handle something with the family business. I didn't back down but I also didn't get loud or disrespectful about it. Just stated my position clearly and stuck to it. He respected me more after that conversation than if I'd just agreed with him to keep peace. You can be a man of honor AND have backbone. They're not mutually exclusive.
Family business stuff is tricky because emotions and money get all tangled up. Sounds like you handled it with your head instead of just your gut.