Step 1: Ionisation isomers are complexes that have the same molecular formula but exchange an anion between the coordination sphere (bonded to the metal) and the position of the counter ion (outside the sphere). They give different ions when dissolved in water.
Step 2: In [CoBr(NH3)5]SO4, bromide is coordinated to cobalt and sulphate is the free counter ion, so in solution it releases SO42-. In [CoSO4(NH3)5]Br, sulphate is coordinated and bromide is the free counter ion, so it releases Br-. The two swap Br- and SO42- between inside and outside the sphere.
Step 3: This exchange of ions between the coordination sphere and the ionisation sphere is the definition of ionisation isomerism (the first gives a white BaSO4 test negative but AgBr positive difference from the second).
Conclusion: Option (iii) is correct. Linkage isomerism needs an ambidentate ligand, coordination isomerism needs both cation and anion to be complex ions, and solvate isomerism involves water molecules inside or outside the sphere; none of these apply here.