First step, is to quit using that HVAC company. Consult your LL Tenant Laws, most states have very specific rules for when/what/how repairs are to be reported and addressed. Also, in most states of which I'm aware do not provide any terms in which rent can be withheld, but rather provide for termination of the agreement if the Landlord does not uphold their requirements. Typically, water and heat are the only items that are considered to be an "emergency" that if unattended could be cause for action agains the Landlord.
If they continue to withhold rent, I would file eviction, if you have everything well documented. As for the HVAC company, as the Owner of the property, if you did not authorize the repairs you could simply refuse to pay the charge or take your lumps, pay the charges and lose their number.
If you're adhering to the LL Tenant Laws and have good documentation you will have the most leverage but I'm not specifically familiar with Florida laws.