How do you disagree with your boss without damaging the relationship?

There's a sequence here, and skipping steps is what damages the relationship, not the disagreement itself.

First, learn your boss's actual interests, not their stated position: what they're measured on, what they're worried about from their own boss, what happens to them if this goes wrong. Most disagreements with a boss fail because the employee argues against a position without knowing the interest underneath it. Second, mirror those interests back before you mention your own idea. 'You're on the hook for the date, and worried this adds risk to that' shows your boss you understand what they're carrying.

This is the whole game: disagreement lands better the more clearly you've proven you understand what you're disagreeing with. Most people skip straight to their counterargument and wonder why it bounces off. Third, steelman their current approach out loud, argue it better than they would. If you can make their case stronger than they can, your disagreement stops looking reflexive and starts looking informed.

Only then offer your idea, framed as something you want them to poke holes in, not something you're defending. Say your ego is out of it and mean it. Leave room for them to add information you don't have.

Fifth, put the idea on paper or in a diagram you both look at, if you can. Criticism lands on the object on the table, not on you, which keeps the relationship intact regardless of which idea wins.