There are times and places where you can't put a name to the people who died. Hitler's Germany, Stalin's Russia, Mao's China, Cambodia. Ironically they all appear to be socialist states, excluding Hitler's Germany Communist states.
In this case though, you're correct. I don't believe that death certificates don't exist for these people, or that said certificates don't state a cause of death. My guess a zealous reporter could debunk these "figures" as John Lott does frequently with false gun stats. But it would take a great deal of time, to do it properly would require examining all deaths in Puerto Rico. I'd be surprised if the beaches were crowed with drowned corpses and the streets with victims of starvation and the media failed to report it at the time, but that's how you find out. I was there in early December 17 which fits this time frame and though the island was a mess, a massive unreported death toll wasn't amongst the things I heard about.
No, you are not correct about how they estimated the 3000. They did it by guessing that 8% of Puerto RIcans left and didbn't come back. In the remaining population there were 16,608 deaths. There was not a massive unreported death toll. There was a 16,608 report death toll. They then took the 8% lower population, and the previous years' mortality rates (same 6 month interval) and calculated that 13,633 should have died.
But the 8% decline in population is just a WAG based on "airline pasenger surveys". There is no indication that those were even barely adequate or whether the people surveyed who said they were leaving for good changed their minds later. The 8% is a junk number. Futhermore the wealthy and healthy left changing the demographics.