I think you are wrong. 'Barbacoa' ... a word used for a platform of some kind in Hispaniola ... appears to be the most likely origin:
"We have to go back to the West Indian island of Hispaniola in the seventeenth century to begin the search for this word. The local Arawakan Indians had a method of erecting a frame of wooden sticks over a fire in order to dry meat. In their language, Taino, they called it a barbacòa, which Spanish explorers borrowed.
This word seems also to have been applied by Europeans to sleeping platforms raised off the ground to reduce the risk of snakebite, presumably without the fire underneath. That extraordinary seaman William Dampier was the first person to use the word in this variant sense, in his New Voyage round the World of 1699: “And lay there all night, upon our Borbecu’s, or frames of Sticks, raised about 3 foot from the Ground”.
It seems that the word began to be applied quite quickly to cooking meat rather than drying it, and for such outdoor cooking to become a social event, even though barbecues in those days differed from the modern suburban ritual in that animals were often cooked whole over pits of hot coals (the Oxford English Dictionary has a lip-smackingly wonderful definition of the process: “To broil or roast (an animal) whole; e.g. to split a hog to the backbone, fill the belly with wine and stuffing, and cook it on a huge gridiron, basting with wine”. The first example known is of the verb, in a work by Aphra Behn of 1690: “Let’s barbicu this fat rogue”, showing that it was well enough known even then to be used figuratively
In 1733, a certain Benjamin Lynde, who lived in Salem, Massachusetts, wrote in his diary “Fair and hot; Browne, barbacue; hack overset”. This is rather a cryptic comment, but then he could hardly have known that nearly three centuries later his jotted note would be transmitted across the Internet as the first ever recorded usage of the noun in our modern sense. It seems that on this hot but pleasant summer’s day he went to some neighbours called Browne to have a barbecue, but that at some point, presumably on the way back, his hack — either his hired horse or carriage — had an accident.
Incidentally, many people believe that barbeque actually derives from the French barbe à queue, that is, “from beard to tail”, signifying the whole of the pig being roasted. Leaving aside the question that pigs don’t have beards (though the allusion would work for goats), the true origin is well authenticated, and the story is just another example of folk etymology."