Mums work opportunities have to fit around the needs of the child - cant your wonderful partner go into work late one day a week? Rearranging working patterns and even entire careers is a fairly common occurence once kids come along.
The child is only four years old. The current arrangements suit her main carer and the child concerned. She needs to spend quality one on one time with her non-resident parent, not be dragged around to be "shown off" to all and sundry.
Why is Mum having one night off a week "selfish" but it's OK for you & your partner to have whole days to tinker with cars, not rearrange your working patterns around the needs of the child, go out most nights whenever you feel like it etc, etc?
Not trying to be rude but given the child's age it's pretty obvious Daddy bailed out and got together with you when DD was only tiny. Her mother has behaved with amazingly good grace regarding contact considering the circumstances. (Not many women can behave so unselfishly when abandoned for another woman while they still have a babe in arms, it's a very vulnerable time).
Now he needs more time off from his responsibility to tinker with cars? red flag alert Mum has been shouldering most of the real responsibility alone for a long time now - why exactly should she give up her one bit of freedom for YOUR benefit?
At only 4 I'm not too convinced that alternate weekends would be that good for the child yet as kids all mature at different rates - perhaps it's something to reconsider when she's a little older? I'm also not convinced you and your partner are quite mature enough to handle that level of responsibility just yet.