Possible Clues:

• Help a pig farmer, we hear, in Caribbean island
• Blue liqueur, or the island where the fruit it's made from grows
• Largest of the ABC islands
• Despicable sort I excluded from a farewell drink
• Mai-tai liqueur
• Island account held by someone despicable (also heartless)
• Thus a drink may make a so well, by the sound of it
• Liqueur of bitter orange peel
• Spirit of the Caribbean
• Drink a round, chosen to replace unknown in junior minister's office