Please don't feel guilty about what's passed, you didn't know what you didn't know and it makes sense to trust your child's school, after all they're the education experts or should be. The school are being ridiculous saying it doesn't matter knowing the why, for several reasons, for her to understand herself, for them to understand how to teach her. If they won't suit their teaching method to what she needs that's their failing. It's pretty poor a school saying it won't change anything no matter what the issue is.
Your plan of action sounds great. A good educational psychologist should be able to point you in the direction of what assessments they think are most appropriate if you outline her issues to them. You may have to travel, it will be worth it for the right specialist. The point about not recognising a word on the next page is an interesting one, and could be an important point. The same with the issues with phonics. It's all evidence of what's going on inside her head.
My DS used to not recognise words, he had the same issue with counting, couldn't seem to recognise the numbers. He loved to swing and I would push him for 30 minutes while we counted together. Sometimes movement helps learning, he's relaxed on the swing too, maybe that made him more receptive. When reading would add sounds to words from letters that weren't there. His spelling was so bad his teacher couldn't mark his work and 20 minutes after writing it he couldn't remember what he'd written. Having got him through it twice now, with counting then reading and now working on writing I still don't know what the issue is. It feels like he can only learn in very specific ways, but I'd be lying if I said I understood why he's had these struggles and how I've been able to resolve them. He's Autistic, but nothing else diagnosed so far.
These are a series of decodable readers you could try, they're designed for children with dyslexia. If you click on a set and then scroll down they have free PDFs you can print off or read on a tablet. https://www.speldsa.org.au/SPELD-SA-Phonic-Readers-New-Series
These are an inexpensive tool that could help. https://www.thedyslexiashop.co.uk/products/crossbow-duo-coloured-reading-rulers
It can feel really overwhelming when you start down this road with possible SEN and all these new ideas and an array of options and decisions that you might feel you don't know enough to make. It does get easier. Be kind to yourself, it's a tough and long process and you need to pace yourself so you can keep going at it and not get burnt out.