Congratulations,
If you are living here in Puerto Plata why not stop in and use the British Consular office based in Beller,
they should be able to point you in the right direction, i.e the process, the passport forms and of course the price.
From what i can remember when i registered my son, i had to go to the JCE office and register my son and then buy the extended birth certificate with the legalized stamp.
At that time the Embassy never asked for the birth certificate to be translated.
Then as with all British passports you need to fill the British passport form out for your child and get someone approved to sign the child's photograph.
Good Luck !!!