As you have stated Florida's State statutes do not expressly require you to provide pest control, although it may fall under the habitability requirement if the landlord's actions contribute (or potentially another tenant in a multi-tenant building) to conditions that foster the pests.
The City's Code as you have discovered apply to the owner of the property. It seems the prior tenants were aware of this when they called code enforcement. It would seem the best option for you is to plan on regular inspection and treatment at your expense, perhaps building the cost into your rental amount.
I believe that depending on the language of your lease the failure of the tenant to keep the property in a condition that discourages the pests and therefore is in violation of a city ordinance, and would likely be a breach of the lease permitting you to request the cure the problem or face a potential termination and potentially eviction.
Good luck, as you know there are always some challanges facing you as a landlord.