My name isXXXXX a licensed attorney. I am honestly sorry for your circumstances, very much appreciate your patronage, and am glad to try and help out.
Accordingly, I'm pleased to say that a solution does exist here, although the procedure has changed fairly recently (Law 191 of 2009, amended June 2010). Here's how this works. You need to address it at the root, meaning through the Puerto Rico Federal Affairs Administration. If I'm understanding your situation correctly, there has been no official name change, so your original birth certificate is correct and reflects the name you want and need to bear. But, the Government denied you access on account of your identification. Well, fortunately the law allows for a number of other folks to obtain your birth certificate (parents or guardians -- if applicable; or if not, then an heir or other authorized person). Here's an overview of the process:
Given your rather unusual circumstances, that's your easiest approach. Otherwise, you're looking at retaining legal counsel, filing a case in Puerto Rico, and going to Court to prove your true identity by means of affidavits, witness statements and so forth.
If you have a follow-up question or need clarification, please just say the word by using "reply" to reach me.
I truly hope all works out for you.