AgCl silver chloride is insoluble, aqua regia will not dissolve but a tiny portion of silver, silver passivates in aqua regia forming an insoluble crust.
In a cell with chlorides in solution, silver at the anode would form a passivated crust protecting the anode from dissolving, some gases can be formed (dangerous chlorine gases), and these bubbles may flake off a little of this crust, just to passivate the next layer, basically you may have some flake off but it would basically be passivated.
Above I said very little silver will dissolve in aqua regia, this is a very minute amount, we sometimes see with our gold when we process it, this silver is not the normal AgCl silver chloride, but silver chloride with a higher oxidation state AgCl2, the AgCl2 you will only have in a concentrated aqua regia solution with high chloride content, when the solution is diluted we get the back that little bit of the silver as the insoluble white powder precipitant of AgCl which is insoluble in the dilute aqua regia.
Basically we can say silver does not dissolve in aqua regia (or at least to enough of an extent we could say that it will).
Silver sulfate is not very soluble, so a sulfate solution will not hold much silver sulfate in the solution, most of the silver would settle to the bottom of the vessel if formed, then we also have the problem of converting the silver sulfate back to a silver metal, making silver sulfate formation undesirable when we do not have to, well that's my opinion.
I would think this type of cell, the silver in solution would also try to plate out with the copper on the cathode (the silver ions that were in solution in the vicinity of the cathode), my guess is that the copper solution would not be clean enough to make a solid plate of copper on the cathode and fluff off to the bottom of the cell, and the silver would form a sulfate, mixing with the silver, yes copper sulfate is more soluble in sulfuric acid but then we still are left with silver sulfate to deal with, if we tried to dissolve the copper leaving silver sulfate (this just doesnt sound like my idea of a good time).