I have cycled through the Keys six times. As far as I know it is safe for anyone on a bicycle---Now. Back in the early eighties it was a different matter. Hostile motorists were there, definitely, and hostile people in general seemed to abound. Things have changed since then. It is safe. The redneck, bike-haters have gone or settled down. The Bubbas have conceded that cyclists have a right to use the roads. Don't worry. As a matter of fact, considering cycling conditions on US Highway One from Maine to Key West, the Keys are probably the safest places on the entire highway, or just about. All I know is I have had no problems there except in 1984 in winter when the motorists were really, seriously unreasonable, even criminal.
That was a long time ago, and in fact, it was kind of like that across the state. Read "Miles From Nowhere" by Barbara Savage. Notice how they were doing all right cycling across the entire US until they got into Florida, and how things really got bad for them in the Keys. I can concur on their perceptions of the Keys and on Florida in general. After all, I was born and raised here, and I have cycled across the state many times, including down in the Keys. That was the past. it has changed.
Certainly you will encounter more traffic in winter, and higher prices. State campgrounds are your best bet, and you might need reservations unless they are on a first-come first-served basis.
Do not be worried about safe cycling. The roads provide plenty or room. Motorists are very reasonable. There are plenty of cyclable sidewalks. There is one long dedicated bike path. It is actually very good for cycling. No problem. Of course, people being what they are, it is possible to have an encounter with a very irate or unreasonable person driving a motor vehicle. However, that can happen anywhere, any time. The big things about the many bridges is they expose you fully to winds. I do not remember having trouble with traffic on any of the bridges.