Does asking my child's opinion undermine my authority?

No, and this is exactly where autonomy support gets confused with permissiveness. They aren't the same thing.

Edward Deci and Richard Ryan's self-determination theory finds that children do better, more motivated and with better wellbeing, when they get a real say that's taken seriously. But a voice is not a veto. Asking what your child thinks shares the reasoning behind a decision; it doesn't transfer the decision itself. You can weigh their input fully and remain the one who decides. The negotiation you're picturing only opens if you've signaled that the hardest push wins, and asking a genuine question is a different move than that.

In practice it runs the other direction. A kid who's only ever been handed rules pushes on them from the outside, because a rule with no reasoning attached is just a wall to lean against.

A kid who's been asked what they think, and heard why the rule is what it is, meets it differently, even the rules that never change. That's the difference, at 11pm, between a kid who understands why the line is there and a kid who only knows there's a line. It isn't entitlement; it's buy-in. What actually breeds entitlement isn't being asked for an opinion. It's being handed the veto you were careful never to offer.

That's its own move, and doing it well has a shape: honoring what they said without giving up the decision. Walk through it step by step here: How to say no after asking your child's opinion.