You might be able to bring suit to correct an erroneous criminal history report as a claim for declaratory or injunctive relief, something that is not barred by governmental immunity.
It is probably, however, necessary to exhaust any administrative remedies to have it corrected outside the court system first. The administrative remedy that is available is described here. It involves filing an Application for Personal Review of Florida Criminal History Record to submit to the Florida Department of Law Enforcement.
You have essentially no chance of overcoming governmental immunity to be awarded money damages for its error, even if you can clearly prove that you suffered money damages as a result of negligence in the maintenance of your criminal history records.
State governments are generally immune from civil liability except where there is a statutory waiver of that protection, in part, due to the 11th Amendment to the United States Constitution (which have been interpreted to affirmatively create government immunity for states despite the fact that its literal language doesn't say so).
Whether Florida has waived liability in this context is unclear. The main relevant Florida statute is Florida Statutes ยง 768.28.
