This advice comes with a big warning. It was ten years ago when I shipped a bike to Puerto Rico, so everything I know may be out of date and just plain wrong.
At the time, I was living on the island. You're right, the prices are much higher there because of lack of competition (each manufacturer's distributor is also a dealer and nearly has a monopoly for that brand). So I bought a lightly used bike in Florida and shipped it to the island, saving myself a good chunk of change and getting a bike that wasn't available locally at the time. I shipped from Jacksonville to San Juan and if I remember correctly the shipping only cost a few hundred dollars, like $300 or $350 or something. Of course that was ten years ago. Some guys in Puerto Rico ship their bikes to Florida just to go to Bike Week in Daytona.
The problem with your plan is that there's an import tax. To title and register your bike in Puerto Rico, you'll have to pay something like a 7% tax. This may have changed in recent years when the island first instituted its current sales tax. When I lived there, there was no sales tax.
There's a lot of paperwork involved, but you can also get people to do that work for you for a minimal fee.
At the time I did this, you could not get your bike beyond the port without registering it and titling it locally. So you're stuck. You have to pay the tax even if you only plan to be there a few months, not permanently.
Personally, given the relatively short amount of time you're going to be in Puerto Rico, I'd rather look for something I could buy on the island. That's assuming the system today is still the same as it was back then. That's a big IF.
Anyone else know better? Boricua maybe?