You Do Need power plug adapter while traveling from Saint Lucia to Saint Barthélemy.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E and in Saint Lucia is G. This is how they look like.

Saint Lucia

In Saint Lucia, outlets of type G are in use.

Type G

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E