KTTNMD, you do not say whether you had the misdemeanor expunged from your criminal records. If the misdemeanor was a first offense, getting an expungement is something that you may wish to consider now, to help restrict any future public access to the conviction.
Unfortunately, having your misdemeanor expunged at this point in time will not help you get your mug shot removed from the site where it appears currently. Your best chance of getting the arrest records site to remove the mug shot is to appeal to them personally, as you have already done.
There was a rather interesting case decided recently by the New Jersey Supreme Court that relates to your concerns here, although in the New Jersey case the records being publicized were ones which had been previously expunged.
In the New Jersey case, "G.D." filed a defamation and invasion of privacy action against Bernard Kenny and the Hudson County Democratic Organization over political flyers that published the fact that G.D. had been convicted of a crime. This crime was one that G.D. had earlier expunged from his criminal records. Under the law and with an expungement, arrests, charges and convictions are deemed not to have occurred.
The New Jersey Supreme Court stated, however, that an expungement "does not transmute a once-true fact into a falsehood. It does not require the excision of records from the historical archives of newspapers or bound volumes of reported decisions or a personal diary. It cannot banish memories. It is not intended to create an Orwellian scheme whereby previously public information - long-maintained in official records - now becomes beyond the reach of public discourse on penalty of a defamation action. Although [New Jersey's] expungement statute generally permits a person whose record has been expunged to misrepresent his past, it doesn't alter the metaphysical truth of his past, nor does it impose a regime of silence on those who know the truth." (see G.D. v Bernard Kenny and the Hudson County Democratic Organization, A-3005-08T3, New Jersey, January 31, 2011)
In Florida, the applicable expungement statutes are found in 943.0585 through 943.059.
The best thing for anyone who wishes to have public access to an arrest or a charge or a conviction limited or restricted is to expunge these from their records as soon as it is legally possible to do so. Once the record has been publicly accessed and published, there is little a person can do to retrieve this information from where it has been published or have it deleted from the publications (although it is possible to have it removed from background check reports).
Good luck.