Used conversion factors to work out the mass of equivalent sulphate for 130.92mg Tl to be 20.5mg sulphate.
You did something wrong here. Mass of Tl is correct, mass of sulfate is not.
Worked out the difference in Cu % and proportioned new molecular weight to be 195.4amu which is three hydrate molecules difference with the old molecular weight so I thought the answer is copper (II) sulphate trihydrate but its dihydrate.
195.4 is OK, show how you got trihydrate from here.
Then tried working out the sulphate percentate of barium sulphate to find mass of sulphate in 116.66mg barium sulphate to be 48mg sulphate.
No, it is not 48 mg.
Note: these calculations are much easier to do in moles. Calculate number of moles of BaSO4 - that gives you immediately number of moles of CuSO
4, water excluded. Calculate mass of CuSO
4 - and you have mass of water. Convert it to moles and you have what you have been looking for - molar ratio water/sulfate.