Sodium Sulfate or Sodium Peroxysulfate?
Ok, let’s take it one step at a time:
Percarbonate is actually not a single molecule, but a double-salt of sodium carbonate and hydrogen peroxide. 
The sodium carbonate reacts with sulfuric acid, resulting in carbon dioxide, sodium sulfate and water. The reaction scheme is: Na2CO3 + H2SO4 -> Na2SO4 + H2O + CO2

For safety reasons I will strongly discourage you from adding carbonate to concentrated sulfuric acid. It will most likely explode in your face! 

But in theory, after having added sufficiently carbonate, Sodium sulfate will precipitate.
But getting a drying effect will be uphill, since water is produced alongside the sulfate, as you can see in the reaction scheme, on top of this is any water that comes from hydrogen peroxide, not to mention the fact that concentrated sulfuric acid is strongly hygroscopic itself.
As for the cleaning of impurities, I fail to see what neither carbonate nor hydrogen peroxide can do. HP can oxidize some organic materials, but most of these will probably already have been destroyed by the sulfuric acid. It can also oxidize some inorganic ions like sulfite (SO3), but do you suspect these to be a problem?
I don’t have detailed knowledge of the conditions needed to form peroxo(mono)sulfuric acid. Undoubtedly some will be formed when HP is added to sulfuric acid, but I am also convinced that only a fraction of the sulfuric acid/HP will react.