I noticed it, but I don’t think the klaxon delay is a bad thing. I think having the menacing music start in that scene works, but like I mentioned before, maybe have it cut out temporarily during the head-turn shot, and see how that feels.
This might be a stretch, but I wonder if something could possibly be put together using Unreal Engine. Using the Phasma model from battlefront, and some kind of console model, you could make a quick insert shot of her pressing a button. Having her activate a silent alarm would make her seem even more intelligent than just sounding a regular alarm, and it might be clearer that it was intentional rather than just a side effect of the shields going down.
Obviously that might be difficult but it might be worth trying. Something like a button underneath the main console, or even a button on the top that blinks after she presses it or something. Hmm, I might do more research on this.
I think not having the klaxons blare during that scene though for the sake of the dialogue being clear is a good reason. And if you don’t think “Lower the shields” would work I don’t think it is a deal breaker. The shields being lowered seems to be implied by the screen behind her anyway.
JEDIT: I think redubbing is a great idea! I think offscreen dialogue is my go-to plan for a lot of my ideas too!! And it’d be a lot easier than animating something.