Is there a way Bungie can get around the "copying" for PS4?
I think there may be a solution but this requires a final update that requires copying. Allocate X amount of space for the game for future updates. This will be the final update done through the PS4 system so bear in mind one more copying. Any future updates can then be done in-game rather than through the PS4 system.
As AdmiralNiko and Lollybomb stated, this isn't on Bungie, this is something on the Playstation platform that Bungie can't just circumnavigate.
If you are experiencing long copy times then you can fix it relatively cheaply since it likely means you're still using the original hard drive in the PS4. If you swap out that hard drive for an SSD a Destiny update will download and copy in under 30 minutes assuming you have decent Internet. I should know because I have a PS4 Pro with an internal SSD and I've never had to wait more than roughly 30 minutes for a Destiny update to download, copy, and install to finish in about about a year now.
No. All code updates have to go through Sony's certification process, and then through their update process. Minor scripting changes could be pushed through in game updates, but anything significant that requires changes to the underlying code cannot.
Oh, and additionally this would require the game to be running to update, meaning that it would not be possible for the update to be downloaded in sleep mode, or while you're doing other things with the system. So for content updates, this would still be an awful idea even if they could do it.