I lived in Puerto Rico for six years when I was young. The traffic in the San Juan area was insane - and that was a while ago. I can't imagine what it might be like now. Also, I'd avoid cycling to Luquillo Beach or El Yunque.
The western half of the island is much quieter. Maybe start a trip out of Mayagüez. The old Ramey Air Force Base (on the NW corner of the island) would be a super area to ride. Still pretty undeveloped with cliffside views of the ocean.