In regards to costs, it's generally the length of proceedings that incurs the higher costs, and it's generally the stubbornness of the parties that incur the longer proceedings. In my own partner's case, for example, he spent $2,000 from beginning to end as a self-represented litigant and offered repeatedly to settle throughout the process, but the other parent refused to negotiate and ended up spending about $15,000 from beginning to end, then settled for what my partner was after anyway.
The other legal aspect I wanted to comment on was the frustration you're undoubtedly feeling about the other parent's actions. Fundamentally, there is nothing you can do about this, and in terms of legal proceedings, it rarely ever works to one parent's favour if they fill their affidavit with statements demonising the other parent. To the contrary, it's better for your partner to show that he is committed to working together with the mother to co-parent the child. Of course, any concerns should be raised, but they should be raised in terms of
the best interests of the child, rather than in terms of
how bad the mother is.
Finally, the care arrangements prior to the severance of contact are quite significant because it will be difficult to argue with those care arrangements that the child doesn't have a meaningful relationship with the father or that it shouldn't be continued. If there's no concerns from either party about whether the relationship between the child and the father is healthy, then it makes the process significantly easier.
Now, personal insight...
It sounds like I probably don't need to say this as you already suspect that it's probably not the right thing to do, but it's extremely important that you never, ever use the child as a messenger, regardless of what the mother has said about communication. Text message communication is all that is necessary between your partner and the other parent, and may even be too generous if it leads to more conflict. It doesn't bode well to suggest in any way that her mother is too poor to take care of her, it would simply be better to say, "Well, you don't need to worry about that, mummy will get you a puffer soon". After that you can either leave it, or send the mother a text informing her that you're either buying the puffer and sending it to her or depositing the money into her account. It's not necessary that you like the mother, but it is necessary that you respect the child's relationship with her, which means acting like mummy is the best mummy in the world whenever the child is present or in earshot.
As to your role as a step-parent, of course it would be wonderful if kids and parents and step-parents could all get along and enjoy their rights freely, but unfortunately, that is an ideal far different from reality, and it's best for the child if you do everything necessary to avoid exacerbating the conflict between her parents. Sometimes, this means not exercising your rights in order to protect the child. If the child is likely to be made to feel bad because she talked to you on the phone, then it's better not to talk to her on the phone. If the mother is likely to have a meltdown in front of the child because you're present at a changeover, then it's better that you don't attend changeovers. Children always need at least one parent doing everything necessary to protect them, even if it means making sacrifices they shouldn't have to make. It's also important that you ensure you're filling the role as a friend, rather than as a parent. It's often the encroachment of a step-parent on strictly parenting matters that prolongs conflict between the parents, contrary to the child's best interests. The court also prefers step-parents to be friends, not parents.
Applying all of this to the legal side of things, the court will order a family report if it feels it is necessary to do so, which is where a family consultant (usually a psychologist) observes and interviews each of the relevant parties, including mother, father, child and step-families. If the family consultant suspects one of the parties is putting the child in the middle or not acting in their best interests, it can detriment a case significantly.
The above comes from my own personal experience as a step-parent, rather than my experience in family law. In addition to that, don't get bogged down by the little tiffs and differences in parenting styles. The child will likely never have privacy during phone conversations and she will likely always hear about how terrible her father is, but all you and your partner can do to counter that impact is to just be good people who never fit into the mold you have been given. It's better to simply accept that hostility exists than it is to try and change one of the parties involved.
Anyway, I hope this helps. I'm not intending to offend, either, more just to help you establish a position that will strengthen your case, rather than risk weakening it.
At the end of the day, what your partner is asking is for his daughter to have the opportunity to continue having a meaningful relationship with him, and without contact, that's simply not possible. The time he is seeking is minimal, and on the facts provided, there's no issues with domestic violence or similar. To me, this case seems simple, but the court may think differently on hearing both sides of the story. In any case, as I said, your partner really has nothing to lose.