The busiest time in Barbados will be from the 11th april when the super 8 series starts. This continues until the 21st and then the next match in Barbados will be the final on th 28th april.
There are some warm up games in Barbados (not at the new kensington oval but at the the 3 W's oval) scheduled for the 5th to 9th march featuring teams from Sri Lanka, Bangldesh, New Zealand and Scotland and it will probably be busy at that time as well but not as busy as April.
If you are going in the middle of march you should be able to get flights and accommodation.
We are going for the first 3 weeks in March and had no problem booikng flights and accommodation (we booked in July and I note that the same deal now is approx £900 dearer) We aslo applied for and received tickets for some of the warm up games