I do the 123 on my daughter, but getting her in the room involves dragging her there kicking and screaming. Then she tries to block the door with her hands and feet making it difficult to close the door and I always end up holding it shut for the time out once I am able to get her in the room (which involves picking her up and moving her away from the door and running to shut it). She won't stay in the room unless I do so. The whole thing is a workout. How do I get around this, or do I? The book doesn't cover this.
The first few times I put my oldest son in his room using this method, I had to put him well into his room, and get quickly out the door and hold it shut, just like you are. The trick then was to not speak or make any noise to indicate that I was still there or listening (though he could of course feel the tension on the door knob). I did not open the door until he settled. After a few times, he stopped trying to leave the room, and before long, he would go to his room and shut the door himself when I said "three."
I would encourage you to persist, because this method worked for our two sons all the way through those tough teen years when there is no way I could have "put" them in their rooms. They knew when I started counting that they had let themselves get out of control, and my "making" them go to their rooms (which involved swearing and yelling and door slamming which I pretended not to hear) was their opportunity to pull it back together. They would emerge 15-30 minutes later when they were calm.
It's very important that you not get into wrestling-type situations with your child, because that stimulates her fight-or-flight response, and then all bets are off. In discipline, please use this rule of thumb: hands off!
I wouldn't use her room as a time-out location. In fact, you might even want to take a break from using the "time-out" as a consequence. I'll give you an example of another sort of consequence that you can use in a step-by-step, escalating way. Let's imagine, as an example, that your daughter normally gets 60 minutes per day to play computer games. If she behaves in an unacceptable way, you explain very calmly what will happen if she misbehaves again: "Missy, you need to be respectful (or whatever) if you want to get your full 60 minutes of your games. Otherwise, I'm going to start taking off minutes. The first time, you'll lose five minutes. The second time, another ten minutes. That would be fifteen minutes off, altogether. The next time, another another fifteen minutes, bringing us up to a total of 30 minutes off. And so on. Do you understand?" It might help to draw a picture of a big clock, and make the chunks of lost time visual.
During this transition period, pick no more than three rules you are going to enforce and stick to them! Make sure that the consequences are not harsh. The two of you need to learn a new way of doing discipline. It will take a bit of time and patience.
If you do use reduced time on the computer as a consequence, you should set up some computer controls ahead of time, with for example Qustodio, which is free and pretty easy to use. You can set how much time per day she may use the computer.
You want the consequences to be graduated (step by step, incremental), and not too harsh. Otherwise, you may be tempted to take back the consequence when you see how upset she is! (I'm speaking from experience here -- my son has been known to break out in hives all over his body from crying.)
Try to stay away from all or nothing consequences.
Along with this structured negative consequence program, you will also need to use the trick of catching her in the act of good behavior and giving her some positive feedback for that. A positive reward might be, "Missy gets to choose what we're having for dinner tomorrow, as a special reward for taking such good care of Kitty, feeding her, giving her clean water, and petting her so nicely."
Edit to make my answer clearer:
How do I get around this, or do I?
You don't, you ditch what's not working, and try something else. This is not a judgment of the 1-2-3 method, nor of you or your daughter. Not every method works for everyone!