Given that Katiebel seems to me to be doing the right thing with carrying out threats etc, it is a tricky one.
I have this slogan of "never let a child think it can be more unreasonable than you can"
She's trying to neutralise your threat capability, and short of increasing the threats to abuse levels, she has got you pinned down.
Thus I'd try striking without warning.
In the situation where you'd threaten sending her to her room, I'd try simply do it.
Even at 3, smart kids have a sense of "law" and "rights". She feels she has a right to a warning and a last chance before sanctions are applied.
She will feel terribly aggrieved by your action (I believe). You are violating her "rights", and the natural order of things.
After you've done this a few times, you can sit down with her, in a clam situation and explain why you are doing this.
In your place, I'd say that because you can't warn her, because of the answering back, you just have to do it, and you are as unhappy about this as she is.
A theme I have with my kids is that my "job" is to help them learn how to be good, and you might find that useful. Yes I know you love her, but they get a lot more "jobs" from TV and books than love. Indeed they get more examples of cookery than love from TV.
Won't work first time of course , because I guess it's a bit of a habit.