What legal options do I have here? I don't think I can sue for defamation of character since the email was sent to me only.
Your rationale about defamation is accurate with respect to the establishment (henceforth "company"). But you may sue the person(s) who approached the company to falsely accuse you of that crime.
Being banned certainly qualifies as special damage (that is, concrete damage), whence you have a viable claim of defamation per quod. If the crime that was falsely imputed to you is a felony or serious crime, then you additionally have a viable claim of defamation per se.
I was thinking that I may have some legal recourse since the email clearly threatens to defame my character if I re-enter the establishment
You have legal remedies, although not necessarily from this angle. The company can credibly argue that it sought to discourage you from contravening the "safety measure" it adopted in response to the accusations made about you.
If it turns out that the company fabricated any false accusations it divulges, though, then you could sue the company for torts related to --and in addition to-- its defamatory falsehoods. So far the information you share here shows no signs of company's involvement in inventing the false accusations.
What legal options do I have here?
You need to ask the company for source and details of the information. In line with this comment, you should also ensure the company is aware of the mistaken identity. If the company declines to listen to you --and ideally see any proofs you have--, that could evidence some sort of tortious conduct on the company's part.
Beware that in Florida a defamed person is required to demand a retraction of the false accusations prior to filing a defamation suit. Absent that request for retraction, it will be very easy for the sued defamer(s) to have your complaint dismissed.
If the company refuses to disclose the source of the false information, then you need to seek injunctive relief in court. That means suing the company so as to (1) compel the company to identify the person(s) who accused you, and perhaps (2) strike the ban that the company put in place as a result of the false accusations.
Even if you don't prevail in striking the ban, the court proceedings would give you the occasion to set the record straight and prevent the company from defaming you if you legitimately expose (to the public) the arbitrariness of its ban. To be clear, the company can always indulge in defaming you for the sake of justifying its ban, although that would be dumb in light of what you will have proved in court by then.
In jurisdictions where a request for retraction is not mandatory, a plaintiff who does not know the identity of his defamer(s) may (1) file suit against "Doe defendants", (2) subpoena the non-party company so as to obtain records related to the false accusations (obviously ensuring that these reveal the authorship thereof), and (3) upon production of subpoena records and requesting the identified defamer for a retraction, amend the complaint to properly identify the defendant. This would be more efficient than filing two suits (one for injunctive relief against the company, and another against the defamers). However, I am uncertain of whether this would work in Florida, given its pre-suit requirement of request for retraction.
