The consideration of the childs welfare will be of primary importance. Not how many nights either parent ticks off.
Usually children of around ten will be asked, and their own thoughts taken into consideration.
Some young children find it quite unsettling to sleep in a 'strange', or 'new' environment, or if the mother has always been the primary carer, some young children may feel insecure without her.
I doubt this will affect your rights as a father in other areas, but in order to reduce stress and upset to a child, I think it's best you respect your daughters view at the moment. She may change her mind, and once familiar with your new surroundings on day visits, she might change her mind.
Don't let your daughter be pulled between the two of you and your estranged wife.
I agree with Sera on this one. It's really that your ex is not forcing your daughter to stay with you because she's not keen. Most children are upset and unsettled when parents split up and that can mean that they like the security of the family home. At this age she is likely to want to in the end, but even if she doesn't that doesn't mean that it's the end of your relationship with her.
If your wife is OK about your daughter sleeping at your place when she is ready then I wouldn't fight it through the courts. It will be expensive and time-consuming and the court would be pretty likely to support you daughter in any case. Just be patient and stay on reasonable terms with your ex: that's the best way to get the overnight stays.