Sorry, but no matter the ex's feelings on your family situation, that should never interfere with your DP seeing his daughter.
If she has refused mediation, then that will be taken into account by the court. They will ask her to give an explanation as to why she didn't want to mediate, and unless she has a really bloody good reason, she will be asked to justify how that that should have any bearing on a child seeing their father.
A note to take forward, as it appears it will go to court because she is refusing mediation, this will then lead to an official contact order, so, if you and your DP want a better/more time arrangement then you will need to go with that proposal at that time.
Also, if she is going to become difficult, also bare in mind that if you ask for any amendments going forward, unless you have extenuating circumstances she is well within her rights to say no, so no ad hoc changes of plan, and you will have to learn to diary your entire lives a year in advance...
As the child becomes older they will have more input, and maybe on the order (if it gets that far) you may want to schedule a mediated review day that she will then have to stick by, otherwise you may find your self stuck in an inflexible order until the age of 16+.
I can't have compassion for the ex, because the child is not her property regardless of her feelings towards your family circumstances. She also doesn't have autonomy over the child, the child has 2 parents who both have valid feelings and opinions.
Who I feel for is the child, the child is always going to become the victim here 😔