I'm sorry the school is giving you grief. I don't know whether it's worth it to fight them and get an IEP, but I do know that no school can provide the highly specific dyslexia training that your child needs. Thank goodness you have the funds to pay for private tutoring! Are you doing OG, or something else? This is going to help your child the most, by far. Observe the lessons, and on days when you're not paying for a session, practice with her, including on weekends. The best remediation is the intensive kind, so that she spends as little time catching up as possible, and can access the curriculum better in a few years.
My kid with an IEP just could not function in school without it. He has dyscalculia, dysgraphia, abysmally low processing speed (double time, and often he still couldn't complete assignments). Also autism and severe inattentive ADHD. He had pull-out instruction in elementary, group speech therapy, a scribe to write for him for a couple of years, and in secondary school, a resource class to finish his work and get organizational help. That's the sort of profile they give IEPs to. In elementary, I retaught at home all the stuff he missed in class, and in secondary, we paid for writing and math tutors, then specific AP/test prep.