Does the HOA's lawyer have a duty or is it standard practice to label
documents (including) as privileged if it contains privileged
While it is sometimes labeled, this is done by attorneys when it is done, mostly to prevent stupid clients from sharing the information and waiving the privilege.
Usually, however, documents are not marked as privileged in advance and when documents are requested from a third-party this analysis is usually done by junior attorneys and paralegals working as a team for the first time.
Formal designation is not required because the definition of what is protected is defined by statute and common law rules on a uniform basis.
The duty to share information with HOA members does not generally extend to privileged information. If a document that would otherwise be required to be disclosed (like attorney invoices) contains privileged information, the usual course of action is to redact the invoice so that it contains only non-privileged information (e.g. a bottom line amount owed and the date of the invoice and the matter).
In an HOA context, the privilege of the HOA as a client belongs functionally to the board and its officers, not to all of the members of the HOA.
Does the Association's contractor who maintains the records have a
duty to act as a gatekeeper to said invoices if they are not labeled?
This depends upon the contract between the Association and the contractor. A well written contract would include this duty, and spell out the mechanics of how it is implemented, but not all contracts live up to best practices.
A related issue is whether a release of privileged information by an independent contractor constitutes a waiver of the privilege by the client if the contractor is not in an agent-principal relationship with the Association, which is often a determination made only after the fact by a court.
Some jurisdictions' rules of civil procedure provide a waiting period between the issuance of a third-party subpoena for documents and the earliest time that those documents can be delivered to a third-party, to allow objections on grounds of privilege or another basis to be raised prior to the disclosure of the documents.
Colorado Rule of Civil Procedure 45, for example, has such a provision. I don't know what the rule is in Florida on that issue.