Unless you are great explorers, taking taxis the entire time is certainly feasible. The Marriott is right in the middle of Palm Beach shops and restaurants, and there's really no need for a rental car. We rented a car the entire week, and didn't regret it because it was handy for grocery runs and exploring the island, but we could have made do with taxis except the day we visited the Donkey Sanctuary. Not sure if we'd have saved any money taking taxis v renting a car, but taxis were certainly convenient and plentiful.
Your strategy is doable, but given that you'd have to return the car by a certain time (which may or may not be convenient to your check-in time at Divi), plus hail a taxi from the airport to Divi Dutch, and then again from Divi Dutch to the airport on your departure day, I'm not sure that it will save you any money. Plus it adds unnecessary hassle IMO.
Personally, I'd either rent a car the entire 9 days or take taxis the entire 9 days and possibly rent a car for day trips.