Thanks to Neill's clue, I was able to do some research. It would appear that the BBC report was not entirely correct.
Section 49 of the Constitution of Jamaica sets out the amending formula for the Constitution, divided into three areas: simple provisions, entrenched provisions and specially entrenched divisions.
Section 34 of the Constitution is one of those specially entrenched provisions. It reads:
There shall be a Parliament of Jamaica which shall consist of Her Majesty, a Senate and a House of Representatives.
For this section to be amended,
The bill, not less than two nor more than six months after its passage through both Houses, has been submitted to the electors qualified to vote for the election of members of the House of Representatives and, on a vote taken in such manner as Parliament may prescribe, the majority of the electors voting have approved the Bill.
Note that it speaks only of a "majority" of electors.
However, if the bill is approved by the House of Representatives but rejected twice by the Senate, it can still be passed,
Not less than two nor more than six months after its rejection by the Senate for the second time, be submitted to the electors qualified to vote for the election of members of the House of Representatives and, if on a vote taken in such manner as Parliament may prescribe, three-fifths of the electors voting approve the Bill, the Bill may be presented to the Governor-General for assent.
In other words, a simple 50%-plus-one approval in a referendum is required to overturn the monarchy. Such a measure would require a 60% approval only where the Senate has twice rejected the bill. Given the pro-republican positions of both leading parties, that would appear unlikely.
So not encouraging news. I hope someone can prove my cursory analysis wrong.