Congratulations! What is your original passport if you're willing to share? Also is it obvious by name/photo that you may not be from Carribean, e.g.: Asian-looking, Li Tsi name.
I have two passports so I have some experience. My original citizenship is Russia and this is quite clear from my surname (think Ivanoff / Petroff). Banks in Europe understand clearly that the CBI passport is not the only one. Banks in other parts of the world - not so much, they don't ask for the original one (I don't volunteer information if not asked). So may be you should look at banks in the parts of the world which are further away from your country of origin.