Yes you need a carseat if you are going to be driving around in a rent a car... I wouldn't take it if all you needed was roundtrip to/from the airport...
PR driving is not all that bad... it is definitely like driving in the US but then again no where else in the world has driving as organized and friendly as the US... And people in the US actually follow the rules... which is not typical of other countries...
We used a rentacar and I needed a carseat for it... But when we rode in a taxi to Old San Juan we didn't need or use one... I just made sure he sat down and buckled him up and I sat next to him... I would tell him your expectations so that your daughter knows what to do without her carseat and that it is temporary bc she's not at home...
You'll be fine without a carseat...