Barbados may begin a process of transition to a republic. The representative of Queen Elizabeth II, Governor General Sandra Mason, announced such a plan in her throne speech in September. Of course, that means it is the government’s program to abolish the monarchy.
An article about this in The Economist mentioned that such plans do not always go smoothly. It cites the case of Trinidad and Tobago, already a republic since 1976, where the head of state (a president selected by parliament) got to “pick the winner” in a situation (1997) that saw two parties tied for the plurality of seats. The author concludes that “fears of a similar confrontation [between president and sitting prime minister] may have led some Caribbean leaders to reconsider their support for republicanism.”
However, there is no necessary reason why the roles of head of state and head of government need to be separate. Nor must it be left to discretion by the head of state when there is an unclear result of the election. These states could adopt something like the Botswana and South Africa models: The parliamentary majority elects a single individual to serve in both roles. Call the person the “president” or the “prime minister” as you wish. But as long as he or she, and the cabinet collectively, depend on confidence of the majority, it is still a parliamentary democracy (albeit maybe not a Westminster system).
In the most recent election (2018) the Barbados Labour Party won all 30 seats. It was a huge win in votes, too, with 72.8%. (In only two of the single-seat districts did Labour win less than 60%.) Still, it would seem that perhaps a more pressing matter might be not the head of state but electoral reform to avoid total sweeps like this.