Sodium Persulfate won't quite rejuventate AP exactly, it will convert Copper (i) Chloride to Copper (ii) Chloride but it won't precipitate any copper out of solution so the saturation would essentially remain the same.
Sodium Sulfide can be used to precipitate Copper out as Copper Sulfide, leaving Sodium Chloride (NaCl aka table salt) in solution. However this doesn't help to rejuvenate the solution as the Chloride ions would then be locked up with Sodium and can't be part of the Copper Chloride etching process.
When Copper Chloride solution is etching metallic copper, it's a redox reaction which goes from Copper (ii) Chloride to Copper (i) Chloride and then back again, so it seems that the way Sodium Persulfate works to enhance it is by maintaining higher levels of Copper (ii) Chloride relative to Copper (i) Chloride. This would speed up etching but the solution would reach saturation more quickly. Sulfate ions would remain in solution but wouldn't affect the process.
I would suggest that if your solution is slowing down, even when you're aerating it, it's more likely that it's loaded with other metals such as Nickel and Iron, rather than just oversaturation with Copper. It should be able to hold up to about 600g of copper per liter in solution, but if it gets too contaminated with other metals it can't continue the etching cycle with Copper so well as it competes for the Chloride ions which are locked up as other metal salts. Even the "cleanest" scrap material such as trimmed RAM fingers will cause this eventually since they have a Nickel layer as well as Copper.
Ferric Chloride (Iron (ii) Chloride) will also etch copper, a bit more slowly than Copper (ii) Chloride, but more likely the Iron in solution will mostly form Ferrous Chloride (Iron (iii) Chloride) which doesn't etch Copper.
Like you I am interested in methods to rejuvenate the etching solution as I have a few gallons which seem to be spent. My experiment with electrolysis produced some weakened but somewhat reusable acid; it didn't clear the HCl completely, and produced quite a lot of Chlorine gas in the process, which would have depleted the HCl, forming water (as well as making quite a stink). I concluded that although it successfully recovers Copper in metallic form, it's not really viable as a way to recover HCl for reuse.
So far the best approach I have come across is to distill off the HCl, and then convert the remaining Copper Chloride solids to Copper Sulfate using dilute Sulfuric acid, so that the Chlorides form HCl with the water, which can then be distilled off again. This will leave dry Copper Sulfate residue which can then be rehydrated and then electrolyzed to recover Sulfuric acid and metallic Copper. There's a good video of this process by Nurdrage
HERE but I haven't tried to do it yet.
While the idea of waste-free regeneration of the solution is an attractive one, I am reluctantly coming round to the conclusion that it can't go on forever and realistically it's probably most economical to eventually just bite the bullet and treat the waste (drop everything out with Iron and then neutralize with lye, to get rust and saltwater) and simply invest in some fresh HCl. In the meantime I'm progressively working through dirtier and dirtier source material to get as much life out of it as I can.
