If you are connecting in JFK and both flights are on one locator there should be a $100 fee for most of B6's destinations. However the B6 website says the following:

JetBlue gladly accepts small cats and dogs in the aircraft cabin on both domestic and international flights (with the exception of flights to Jamaica, Barbados, and Saint Lucia). To book your pet, call 1-800-JETBLUE (538-2583). Please note: Pets cannot be booked online.

Based on this wording there there might be an additional fee for St. Lucia or they may not allow you to fly with your pet at all!!! Call back and get a supervisor to clarify and confirm that you will be allowed to take your pet when departing from St. Lucia and in addition get something emailed to you confirming that you will be allowed to depart from St. Lucia with your pet.

Also I would double check the weight for both your pet and the bag as some GA's might be really limit you to no more than 20 pounds combined.

Also have all of your up to date vet paperwork in order for this trip.

Required pet vaccination and documentation for each destination of your trip
Identification tags and vet certificate
A carrier that does not exceed 17"L x 12.5"W x 8.5"H. Your pet must be able to stand up and move around in the carrier with ease

Combined weight of your pet and carrier may not exceed 20 pounds