Peaceful Liz, can I be blunt? (I am going to be, and if it isn't helpful, just ignore me, I won't be at all offended).
If your DD has dyslexia, then whether you get an Ed Psych to say it or not, she will still have it.
All resisting assessment will do, in my opinon, is allow your DD to think that she is just 'no good' at this stuff.
If your DD has dyslexia, it can actually be really very empowering. She will know there is a reason why things are tough for her, and there are things that will help.
My DD1 has significant special needs. I have had to push and push and push to keep people interested in her needs and her progress. It isn't that they don't want to, it is that the demand for intervention is so high, and the workers so few, that they will put off anyone who isn't determined. DD1 will start a Special school in September, which has only been allocated because I went over the head of the person who was meant to be organising her assessment and pushed for assessment myself (I did tell her I was going to - nicely ).
Your DD has been given 10 minutes of 1:1 support - what on earth can that achieve?? By the time she has had a couple of minutes to warm up and focus, 5 minutes to 'do stuff' and a couple of minutes to process what she has been doing, it is over.
She has spent 3 years with very little support, and is now very behind.
You want her to work independently. And she can. But first she needs significant help to build the foundations of confidence and knowledge. Trying to make her run when she can't walk is pointless and heartbreaking - for her and for you.
If I were you, I would go to the SENCO, and insist that she puts her on the Ed Psych waiting list. If she says no, tell her that if she doesn't, you will apply for a Statutory Assessment of Special Educational Needs - then see how fast she changes her mind!!!!