Parochial schools are not obligated to offer special needs children the supports and interventions they may require. Public schools do operate under that obligation - if you take public money you must comply with public laws. It all depends on the situation, some children are able to succeed with fewer supports, others are not.
I would agree that a neuropsychologist evaluation of your daughter would provide many of the answers you seek - of course, could also open many new questions, but if ultimately she gets what she needs to succeed in life, that's all that really matters in the end.
Neuropsychs can take MONTHS to see due to backlog, so if it was my child, I'd call the elementary school district to which she is assigned (even though she attends a private school, she is still their responsibility legally) and ask for them to open a case study evaluation (an IEP evaluation) due to the concerns you mention above. That might prove a quicker solution and would not cost you or your insurance a cent.
Good luck!