You should consult this web page from the Florida government. First, you are expected to report the injury to the employer within 30 days, and if you do not, your claim may be denied. You have actually done that (they have actual knowledge of the injury so a future petition is not barred by the 30 day rule). Then within 7 days they are to report the injury to their insurance company. They acknowledge that employers may refuse to report injuries, and clarify that the employee has a right to report the injury to the insurance company (they offer assistance in doing that). Your medical expenses would be covered. There is also a provision that you can get compensation for lost time.
It is against the law for them to fire you for filing a claim or attempting to file a claim ("No employer shall discharge, threaten to discharge, intimidate, or coerce any employee by reason of such employee’s valid claim for compensation or attempt to claim compensation under the Workers’ Compensation Law").
Regardless of the effect on the location's record of reported accidents or the amount of paperwork involved, it is not obviously in the interest of an employee to forgo their rights to compensation. The employee's attorney can advise the employee whether the company's alternative offer is advantageous. If one does not take action (notifying the insurance company for example), eventually they will have waived your rights for compensation.
Supposing that the employee files a report with the insurance company, and the company then fires her in retaliation. This would not involve any unemployment compensation. There is no provision under the law for the state to litigate on her behalf, although the attorney general might file a class action suit if the company is engaged in widespread illegal terminations. So it would fall on the employee to litigate the matter: retaliation for filing a workman's comp claim is one of the recognized grounds for a wrongful termination suit. There are numerous attorneys who may take such a case on a contingency basis (it would depend on the actual details, to be discussed with the attorney).