I would be less concerned with a school "messing up" an LD eval than I would be concerned you might not get as much information through a school eval. None of us can predict reasonably how an eval will go at your specific school (unless we've been through an eval there) - I would suggest you try to network through as many parents locally as you can who've had experience with LD evals through your school and your school district. If you have an advocate's group you can ask there too.
Re the differences between school district and private - for kids who are being tested for LDs - it's been our experience that you receive more testing through a private neuropsych - school testing will show how the LDs impact different types of tests, but the neuropsych will test one step further to determine what is at the root of those differences. For instance, for my ds who is dysgraphic and has an expressive language disorder - the eval he went through at school to qualify for eligibility for an IEP under SLD/written expression included ability and achievement testing (WJ-III for both) and the TOWL (Test of Written Language). Work samples were also reviewed, as well as state testing results. Grades would have been reviewed if he'd been at a school that gave grades. If we'd had no other testing what we would have learned from that was: ds has a challenge with writing. It was also suggested that some of the issues with writing could also be the result of many other things, possibly ADHD, possibly lack of motivation, etc.
The neurospych testing included all of the above (WISC for IQ instead of WJ-III Cog), *plus* - detailed parent interview (birth through present), behavioral surveys from parents and teachers, executive functioning tests, attention tests, fine motor and visual integration tests. What we were able to learn from the neurospych was very specific - we came away with a diagnosis, with an understanding of *what* we were dealing with, with a suggested plan forward not just for one year but through college (general plan, not detailed at that point in time), a review of which schools in our district would best fit ds' needs, recommendations for private therapy providers, recommendations for what we could expect the school to provide in terms of remediation and accommodations, and a resource that we could come back to at future dates to ask very specific questions of re our child.
So my experience has been that having that neurospych info was *extremely* important to us in understanding how to approach our ds' education as well as understanding how his disabilities impact his life outside of school.
One other small thing that can also really matter - the players on my ds' school "team" were constantly changing from year to year. We had three different school psychs over three years, two different SPED reps, different teachers each year. Our neuropsych is still easily accessible years after ds' original diagnosis - she didn't disappear into the ether like the school psychs seem to

Hope some of that helps!
polarbear