I agree with Enid -- once it had got to that stage I would have taken her swimming costume off (interpreting the fact that she wanted me to take it off as a step in the right direction, even if she wasn't expressing it the right way) and worried about the lack of respect later. And I agree with SecurMummy that you may be trying to explain too much and offer too many choices. There's a limit to what a six-year-old can take in.
Personally I'd have given her initial empathy and offered an alternative, but once the flinging-on-floor started would have ignored her completely and probably left the room. Depending on whether this was a pattern of behaviour I might have identified some longer-term consequences later on.
If adopting your time-out approach I'd have told DD1 (or DH) that I was sorry about their homework/need to rest but that this was important, so could they please watch the baby for a little while, then put DD2 in her room (etc.) and held the door shut for as long as it took.
What happened was that you did the initial empathy/alternatives, she carried on tantruming, you tried ignoring but didn't keep it up and kept trying to reason with her (she wins), you tried time out but only kept it up for ten minutes (she wins), and finally lost it enough to smack her (she wins again). There doesn't sound to have been any real consistency.
It sounds as though you are frazzled and low on the reserves of emotional energy you need to follow through with the non-smacking approaches. Does your DH work nights or shifts?