1. Yes, you can ship your vehicle to Barbados. It’s very much legal and allowed and is widely practiced. Just last year my uncle shipped in a Chevrolet which is left hand drive.
2. Yes, you can sell it before you leave.
What you would need to do is to decide whether or not it is worth it to ship a car to Barbados. Cars accrue immense levels of duties by HM Customs Barbados. For example, my uncle spent $15,000 USD in accumulated taxes and duties to import a $8,500 USD car and that had nothing to do with the shipping cost. IMHO, that was not worth it since he only drives it a couple times a year.
What I would suggest is that you look at the cost of purchasing a car here off the local market compared to shipping yours in. The reason for that is that left hand drive vehicles are not that popular in Barbados so you may find difficulty in having it sold.
Have a look at Nassco Ltd – (Toyota, Lexus etc), Simpson Motors (Mitsubishi, Chevrolet, Suzuki, Mercedes, Porsche etc), MQI (Madza, BMW), Courtesy Garage (Nissan, Fiat, Hyundai etc).
Just type the company names into Google and they will come up.
Hope this helps