I took dd to visit family in Barbados when she was 6 months old, with no problems. Barbados is a very 'easy' place to visit, family-friendly, good medical facilities. it is a very well developed holiday destination and I'm sure you will be able to book minicabs with car seats etc.
I can't help on hotels, as we stayed with family, but I think you do need to book a ticket for your baby, even though they'll be on your lap. You can try to book a bassinet (there are never enough for all the babies on board, IME) but be warned: they are very small and even at 4mo your baby might find them a squeeze and prefer to be on your lap. It is an 8 hour plane flight.
In Barbados, the west coast is considered the most desirable, but the south coast generally more affordable. The east coast is less developed for tourists, but beautiful. It's a small island (about 20 miles by six, I think) and you can hire a car and tour round. It's a very safe and friendly country with a low crime rate.
Slightly bemused by your parents' reasoning, though. What food do you need for weaning that you can't get in the Caribbean? In fact, I started weaning while there - bought baby rice locally, added purees of local squash, papaya, mango etc. Only thing she didn't like was cho cho, but that just shows her good sense.