Based on my recent experience renting in Florida out of Orlando and Tampa area, not many FBOs will allow you to fly alone with their aircraft to the Bahamas. Don't know why things have changed.
For flying to the Bahamas, you might want to check this AOPA link for some valuable information.
Flying to Bahamas - AOPA
To retain an aircraft for a longer period without flying back to aircraft home base, you will have to fly a minimum of 2-3 hours a day, depending on rental policies.
And, don't take any risk to rent without adeqaute insurance coverage. You can get insurance for about $150. It is worth it though.
WP