I am sorry it is all happening. Is it even legal to withdraw an offer - presumably you have already paid the deposit, so entered into a contract with the school? However, even if you can stay - probably best for all to find another solution.
Keep trying other private schools. Often places do open up as time goes on. DD1's "impossible to get into" school at assessment time had 2 spots in Reception as kids didn't show up in Sept. And people also moved during the year, so there were places here and there. Same in other private schools in the area.
In the meanwhile, I think you need to become more realistic about your DD's behaviour. You must realise that other parents do not want their kids to be subjected to a disruptive child in a school setting.
DD1, currently in Y1, has a girl in her class who is constantly in trouble. Luckily, there was no biting but in Reception she was into pinching. Parents are also of the attitude of "striving to establish boundaries" but quietly think that their daughter is "not that bad" and people are overreacting. That attitude helps no one, least of all their daughter. She barely has any friends and has a firm reputation of being the troublemaker.
Your daughter is still small and there is still time to fix whatever issues that are leading to her behaviour. At nearly 4yo biting repeatedly and kicking strangers is not OK. And, in my experience, it is normally not the child's fault, but comes from no consistency with discipline/consequences from bad behaviour at home.
Both of my DDs went through a boundaries testing phase and during that period spend some time crying in a "naughty corner". They don't always get along now, but know very firmly that hitting, etc is completely unacceptable in our house. And that misbehaviour ALWAYS leads to consequences, not just threats.
BTW birth of a sibling is only bound to make it all a lot worse, unfortunately. So, if you still have time - you need to start now.