"First, obtain a small claims judgment for the value of the carpet. Second, file for immediate eviction if the judgment is not paid'"
You can not file for immediate eviction because the tenant did not pay a judgement. A seperate small claims judgement can not be enforced under the lease. What would be the violation or grounds? The landlord would still have to serve the tenant with proper notice to cure as per Florida law. As long as the tenant still occupied the rental, the court would say this is a security deposit action. The proper course of action is to follow Florida state law in recovering damages from the tenant under the Landlord Tenant Act. Also if the landlord has won a judgement for the carpet, the landlord would not be able to apply the security deposit towards the carpet damage when the tenant vacated.
"Been there. Done that" Well in Florida, you might wind up paying the tenant money for such actions.