Congratulations.
A few things….
Firstly, why won’t you be getting money from her dad? If your ex is earning, then he is required to contribute to the support of his daughter. It isn’t his choice, you are entitled to put in a CMS claim and you don’t need him to be on the birth certificate.
Secondly, it’s better to give your daughter your surname if you are going to be her primary carer.
If her surname is different, you’ll spend 18 years being called the wrong name by doctors, dentists and teachers. You’ll waste hours explaining that your surname is different. And you’ll have to prove your daughter IS your daughter to border agencies and immigration. In some places you’ll need a letter of consent from your absent ex even to take your daughter on holiday.
If you leave his name off the birth certificate, he can apply to be added but he has to make the effort.
You’ll get child benefit plus UC on the basis of need I guess, including to support your child. Add her details as soon as possible to get the claim underway. Remember to claim Child benefit immediately. Child Maintenance payments don’t affect UC so anything your ex contributes is extra.
Good luck.