This is going to be a long post because it's something I feel really strongly about.
The "good" thing about it is that you have hard evidence. You have the fact she wasn't invited to the party and the screen shot of the texts and the screen shot of the FaceTime attempts. Those can't be denied. The contact was malicious because otherwise she would be at the party. The evidence shows that she didn't respond to the texts or pick up the FaceTime calls. So it shows harassment. The fact that you are not complaining about the taunting before the party, or her lack of invitation puts you in a very strong position, because what you are complaining about is evidenced.
If you didn't want to speak to the mother directly and you spoke to school, they can definitely deal with this. As a teacher, (used to be head of year) I would speak to your DD and create a plan. She needs to be in charge of it because otherwise she feels things are being "done" to her. And, after all, we want things better for her and for her to have strength in the situation.
The most popular route when I have dealt with things like this is that I call in G to say that a few girls had been to see me, very upset that they felt she was being mean to DD. I would say that I hadn't even spoken to DD yet but that there was something about text messages and FaceTimes going on at some party. Could she elaborate?
She would then be caught on the back foot. Which of her cronies was not supportive of her? There were "a few" as well, so they must have badmouthed her. Her little gang isn't quite as tight as she thought. If she asks them all individually, it might not be them but then they would know there was descension in the ranks and the whole group could topple. If she doesn't mention it, who can she really trust? The main 2 things that bullies want - power and support have been questioned. For this reason she is unlikely to ask the cronies about who "grassed" but if she does, they can all legitimately deny it. Chances are they will start pulling away from her and creating support for each other.
Then I ask you for the screen shots. Call G back in again. G, the problem is bigger than I thought. It sounds so mean. I asked DD's mum to look at the phone and send in the screenshots, on the back of the reports I heard from the girls. Did you know it's an offence to send texts that cause alarm or distress? Can you read this out loud to me? I'm not sure if we should ask your mum to come in for a parent conference. I'm going to need to seek advice, can you sit outside for a moment.
2 routes here - either tell G that you are going to be checking in regularly with the girls that came to report it, as well as her, and also DD ("even though I haven't even spoken to her about it yet") and that the tiniest whiff of nastiness you will be discussing this with the headteacher and recommending an exclusion. Or, route 2, phone G's mother "Hello Mrs G'sMum, school here. We've got a problem here. I have had some girls come to see me this morning, to report some nasty behaviour that has been going on involving G - no I'm afraid I never give out names of those reporting issues, just like if G came to see me about something she saw, I wouldn't give out her name to another parent, I'm sure you understand - in any case, yes, I appreciate there was a party and she didn't invite DD. Absolutely fine - she's within her rights to ask who she wants - however the girls said that they felt awful that G was sending some really nasty messages to DD. I had to ask DD's mum to take screen shots and send them in. Mrs G'smum I have to say, I think you would be so upset and angry if you saw them. You have really high standards for her I know. This falls far short, trust me. You want to come in? Absolutely - I think you need to see this. Mrs DD'smum said that DD was extremely upset over this, and as we both know, communications that cause alarm or distress are illegal. Do you think it would be helpful if we asked in our beat officer to talk to G with you? No? You feel you can deal with this yourself? Ok. I'll make a note of it that you will be dealing with it really firmly at home, and hold off on that for the moment. What I will be doing, is to monitor this really really closely. And I promise that if there is the tiniest whiff of further problems I will give you a call straight away. I have told G that if things continue after this intervention she could face an exclusion. We have been given very first guidelines from the government about persistent bullying behaviour. No no, I'm not saying G is a bully. I'm saying she is engaging in bullying behaviour and if she does not stop, straightaway, people will draw that conclusion, and that's a really hard thing to shake off."
There are lots of ways to skin a rabbit but while there was breath in my body I would be calling her out on it. G expects secrecy and nobody daring to question her. Do exactly that, whether through the mother or through the school. I promise this is squashable.