What you are paying for is an enhanced background check by a private company. By law, private companies do not have access to Interpol databases, but in the world︀ of spooks you never really know.
Separately from the background checks, there might well be︁ an Interpol check, but that is not actually what you are paying for. An Interpol︂ check does not necessarily (but could) notify the state.
It is not strictly true that︃ the local police certificate is "enough". It is part of the big picture. If we︄ imagine I am the person responsible for the check, I am looking at a range︅ of factors and writing up a report based on firm evidence that will then be︆ presented to a committee. I am not necessarily going to follow the exact same procedure︇ with all cases, because that would be too predictable and open to manipulation, and circumstances︈ may be different. So there is no right or wrong answer - it is flexible.︉ This is where I am saying you have to sit down with the Caribbean government︊ and explain the special circumstances, and basically ask them to be especially discreet in your︋ case with the country of origin.
If your agent is telling you that you are︌ paying for an Interpol check that naturally notifies the state of origin, I suspect your︍ agent is out of his depth, as most would be in this case. I would︎ recommend you choose a different agent. By all means tell them everything before you pay,️ but don't realistically expect them to give you concrete answers before you pay, because this is serious work, not something an agent can be expected to do without a firm financial commitment.