I can understand how you would feel, when trying to help and offer support. Seeing the school offer all this support and wanting an open discussion with her, but having her shut down, disengage and withdraw from any discussion. I do totally get how as a loving mother, you’d become frustrated as you just want her to be okay and from your point of view, everyone is giving her everything she needs, but she’s not willing to engage.
It’s hard, but you have to try and see things differently. She isn’t actively choosing not to engage, or not to help herself, or to ignore you. Anxiety is fear that is there to protect us. As much as anxiety is debilitating, it also makes you feel safe. When she’s silent, stuck in her room, not going to school, she feels safe. I agree with counselling being better than CBT, as you might get to the bottom of why she feels unsafe. If she doesn’t know anything about a laptop and hasn’t asked for one and wouldn’t realise whether you gave it or not, then don’t give it if you think it will keep her even more withdrawn. Although withholding it as some sort of punishment wouldn’t work, if she never knew she was getting one anyway.
If she’s asked for one or is expecting one and you withhold it, that could make things worse. Imagine that you feel unsafe doing something so you shut yourself away to protect yourself, then your own mother holds something over your head, giving you an ultimatum that you must go to these unsafe places and have the unsafe conversations or else you won’t get what you’ve asked for. She’ll probably go more into herself.
Withholding her laptop isn’t suddenly going to make her feel safe enough to open up to you and to go to school. All it’s going to do is damage your relationship and her perception of you. Have you tried writing her a letter telling her how much you love her and that you want to know how she’s feeling? Ask for her to write a letter back, or draw how she feels. Perhaps if she thinks a conversation with you will just be around school and her having to go in, then she doesn’t want to engage with that kind of conversation as she doesn’t want to go into school? Conversations around how she feels generally, might help. Writing a letter and sliding it under her door is less confrontational than coming in her room and trying to start up a conversation.
Sorry, that was long-winded and I’m by no means an expert, but just thinking out loud and trying to help. Shortened version - if she suspects or expects that she’s getting a laptop for her birthday, then give her the laptop.