If care is 50/50 there should be no discussion about child maintenance. In relation to benefits it is a very sticky situation and we’ll worth speaking to your local citizens advice. Also maybe worth agreeing that any child benefit payments get directly allocated to your daughter. Everyone’s situation is different and we don’t know a lot about your financial situation or if you are able to get another council property in your own right so worth a chat with your local authority housing team.
Good luck and keep your head up