Dominion fleets and a West Indies federation

If the dominions had battlecruisers, would Caribbean colonies such as Jamaica or Barbados receive ships, and what kinds?

Sloops or destroyers are considered affordable, perhaps with a destroyer leader as flagship; larger named or donated ships would serve in the Royal Navy.

The hypothetical Caribbean fleets are constrained by the colonies' economies and populations. Sloops are the principal suggestion, with destroyers also possible and a destroyer leader proposed as a flagship. Cruisers named for Caribbean colonies would still be Royal Navy ships. A colony could also pay for a larger ship without supplying its crew, following the comparison with HMS New Zealand. Local ownership, financial contribution, and responsibility for manning therefore produce different answers to the question.