I would suggest trying to avoid court if you can.
Have you been to mediation. (you have to go their first before court anyway)
Try and work something out with the mother so that your DP gets more time with his DD.
You can also do child inclusive mediation and the SD can talk to a mediator on her own to have her voice heard.
If thats what the SD really wants then the mother might listen to the mediatiors over you guys.
Court is a long and tedious process and might not get you the result you want in the end.
Perhaps you could offer the mother different arrangments and see what suits her best.
-Your DP could offer to do school pick up a few nights a week and have her over for dinner.
-More time in school holidays
-EOW and 2 nights during the week