The IUPAC name for Ba[BrF₄]₂ is

barium tetrafluorobromate (III)

The correct answer is (4) barium tetrafluorobromate (III).

The IUPAC name for a coordination compound is based on the following rules:

The ligand name is given first, followed by the name of the metal ion.

If the ligand is a polydentate ligand, the prefix "bis", "tris", etc. is used to indicate the number of donor atoms in the ligand.

The oxidation state of the metal ion is given in parentheses after the metal name.

In the case of the complex Ba[BrF₄]₂, the ligand is tetrafluorobromate, which is a polydentate ligand with 4 donor atoms. The oxidation state of barium in the complex is +2. So, the IUPAC name of the complex is barium tetrafluorobromate (III).

The other options are incorrect. Option (1), barium bromofluorite (III), is incorrect because the ligand is not called "bromofluorite". Option (2), bis (tetrafluorobromium) barium (II), is incorrect because the ligand is not called "tetrafluorobromium". Option (3), barium bis (tetrafluorobromate) (V), is incorrect because the oxidation state of barium is +2, not +5.