1. Identify the ligands and oxidation state

   The complex is $[\mathrm{PtBr}_2(\mathrm{PMe}_3)_2]$

   • $\mathrm{Br}^-$ is the ligand bromo.
   • $\mathrm{PMe}_3$ is the neutral ligand trimethylphosphine.
   • Since the complex is neutral: $x + 2(-1) + 2(0) = 0$ $x - 2 = 0 \Rightarrow x = +2$ So the metal is platinum(II).

2. Count identical ligands

   • Two bromo ligands $\Rightarrow$ dibromo
   • Two trimethylphosphine ligands $\Rightarrow$ bis(trimethylphosphine) or di(trimethylphosphine)?

   For ligands whose names already contain a prefix or are more complex names, IUPAC uses bis, tris, etc. Since trimethylphosphine is a substituted ligand name, we use: $\text{bis(trimethylphosphine)}$

3. Arrange ligand names alphabetically

   Ligands are listed alphabetically ignoring multiplicative prefixes.

   • bromo
   • trimethylphosphine

   Since $b$ comes before $t$, the order is: $\text{dibromo bis(trimethylphosphine)}$

4. Write the full name

   Therefore, the correct IUPAC name is: $\boxed{\text{dibromobis(trimethylphosphine)platinum(II)}}$

5. Check options

   • A: bis(trimethylphosphine)dibromoplatinum(II)
     Incorrect: ligands are not in alphabetical order.

   • B: dibromodi(trimethylphosphine)platinum(II)
     Incorrect: for trimethylphosphine, bis should be used instead of di.

   • C: dibromobis(trimethylphosphine)platinum(II)
     Correct.

   • D: bis[bromo(trimethylphosphine)]platinum(II)
     Incorrect: this suggests a different grouping of ligands.

Hence, the correct option is C.