What does the parenting plan state? Does it include a provision about the mother collecting the child on Mondays for tutoring? Do you know who the tutor is and where to contact them?
I can't predict what the court will hear or accept, but if you have 50/50 care, then court orders maybe aren't necessary. If the conflict is too high, the court won't often order equal time - you might end up with more or less than what you currently have, so you might consider some other options first. Realistically, the court would likely have no issue with you taking the child to tutoring (it would be healthy and even encouraged to be a part of that aspect of the child's life), but it may have issue with the level of conflict and whether either party has made efforts to reduce that conflict, so there may be unintended consequences associated with seeking a court order.
Definitely, the mother's criticism is a lot to deal with, but there are other ways you can deal with that, such as blocking her number and moving to a different communication medium, such as e-mail or a communication book. The problem with texts and phone calls is that emotions can come to life straight away, whereas e-mail or a communication book means there is a time gap in which the anger can die down a bit. Other things you can do include not responding to texts criticising your parenting or personal life, and only responding to text messages that are of relevance to the child, such as changes to the care schedule or school meetings, etc. If you respond to text messages that are about your personal life, you're just feeding the beast, and it's better and healthier to rise above that petty tug-o'-war.
It's universally in a child's best interests not to be exposed to conflict, but if the mother isn't willing to try and reduce the conflict, that job is left solely to you. Unfortunate, yes, but quite often, it's found that if your behaviour changes, the other party's will very slowly start to change, too. Relationships Australia offers a free post-separation parenting course that is very helpful in teaching communication styles to reduce conflict. One of the main things taught is about 'the business of parenting', which is where you communicate with the other party as though it is a business relationship, like a client or a supplier. That means being respectful and polite (since you need that business relationship to be workable), but also not entering into personal conversations (the same as you would not do with a business colleague).
Anyway, this isn't really a post offering guidance for the legal aspect, but more an informative one from my own experience as a step-mother entering into a life with my husband and his high-conflict ex-wife and co-parent. Like you, we have one unintended changeover during the child's time with us so that she can attend an extra-curricular activity, and like you, it was often followed by derogatory and hateful commentary by the mother about our personal lives and parenting styles. Now, however, that aspect has died down, and I believe it's because we just didn't feed the beast, no matter how tempting it was to do so.
I hope this in some way helps. I know that dealing with a hostile co-parent can be draining, but that just means the co-parent is still dealing with some demons likely related to the breakdown of the former relationship, and there is nothing you can do or do to improve that situation for her. However, you don't have to be dragged down by it. Your house, your rules, her house, her rules, end of story.
