I'm afraid a lot of the advice on this thread is incorrect.
Your MP can't help. There is no point going to them.
Similarly, there is no point going straight to the head of children's services or anyone else at the LA unless you have clear evidence that a mistake has been made. They can't help either.
Contrary to what several people have said on here, you do not have to show that the Admissions Code has been breached or that a mistake has been made in order to win an appeal. If either of those things is true you have a very strong case but it is also possible to win a secondary school appeal by showing that the disadvantage to your child from not being admitted to this school outweighs any problems the school will face from having to cope with an additional pupil.
Note that this is about disadvantage to your child, not problems for you. The appeal panel may be sympathetic to the problems a parent may face in getting children to schools that are miles apart but you will not win an appeal on that basis. To win on the basis of children being sent to separate schools you would need professional evidence to show that your children have a stronger need than others to be at the same school.
Note that you are appealing for the school you want, not against the school you've been offered. The allocated school's reputation for bullying won't help. If your child goes there, is bullied and the school fail to take effective action that would be something you could use at appeal but at this stage the appeal panel is going to assume that your child won't be bullied.
You need to show that the school you want offers things that are particularly relevant to your child that they would miss out on if they go to the allocated school. Do they offer subjects that aren't available at the allocated school? What about extra-curricular activities? You need to identify what is different about this school and relate it to your child. So, for example, if your child is musically talented and the school you want has lots of extra-curricular musical activities, that is very relevant.
As one person up thread has said, you can appeal for as many schools as you want. You are likely to get a different appeal panel for each school so more appeals increases your chances that one will succeed.
Regarding the waiting list, it is in operation now but there won't be any movement yet. You should be on the waiting list for any school that you named as a higher preference than the school you got. Make sure that is the case. Some LAs limit the number of waiting lists you can be on, others allow you to be on as many waiting lists as you want. If you want to go on waiting lists for schools you didn't apply for you will need to check your council's policy on this. They may not want to tell you your position on the waiting list and it is unlikely they will want to speculate about how much movement there will be.
If you give more details of your case and any arguments you might want to use in your appeal, I and other admission experts on here will be happy to help.
Finally, I know you don't want your daughter to go there but you should accept the place you've been offered. If you don't the LA is not under any obligation to come up with another place for her. Rejecting the place will not move you up any waiting lists and it won't help you at appeal - indeed, it could damage your case at appeal.