I think that most 13 years olds feel sad/confused/angry due to the hormones and spend a lot of time trying to figure out why.
I suspect the lies she told the counsellor are the things she feels rather than things you have said/done. The fact that she told the counsellor to tell you makes me think that they are topics that need to be discussed.
This is not on the same level, but my DD is struggling with lockdown and she (although younger) needs a safe space to be negative. To talk about how rubbish it is and how bad she feels.
She also has questions, some of them require grown up answers, but I've realised she needs more explanation then I had previously thought. It's taking time and effort but I think we are, in bits and pieces, working our way through the problems together by allowing her to be negative and to talk about the hard stuff.
The divorce will have involved events that only you know about, but your DD won't understand some of the things that relate to her, the new girlfriend and her status.
She may need to understand that you were hurt by the friend (she's old enough to understand that), and that you were very upset, and talk about how your feelings have changed over time.
She might need that in order to give herself permission to get along with the girlfriend.
Kids know that adults feel sadness and pain, and I think they need to know that those feelings change over time.
Learning to sit with sadness and pain is an important life skill, so is realising that there is a way out of sadness, even if the reason for the sadness (divorce) cannot be fixed.
Keep up with the counselling, maybe get some for yourself, and talk to her about the topics she raises with her counsellor.
You haven't made her unhappy, lots of kids aren't happy at that age, and sending her away us not the right thing to do.