Also be aware that they are a game animal in Florida, specifically a Fur-bearer, so they have some protections under the Fish & Game regulations.
I'm not aware of any provisions in Florida law that allows one to kill a game animal that may be threatening a dog.
Florida law provide the use of force to protect people, not pets.
There are also laws (new last year) that may restrict shooting in a residential area:
In my experience coons are afraid of people and dogs.. . .who recklessly or negligently discharges a firearm outdoors on any property used primarily as the site of a dwelling as defined in s. 776.013 or zoned exclusively for residential use commits a misdemeanor of the first degree. . .
Also note you can obtain a hunting license and hunt them year-round.