Around the time when destiny came out, there was a limited time offer for people to get Destiny on the next gen consoles, along with a free DLC transfer. So far as I have seen, that was only for people who got the digital version of Destiny for the PS3 or 360. This happened around January 2015. The only way to get DLC from the PS3 to PS4 is if it is cross-platform.
I hope this helped, if you have more questions about Destiny on the PS4 ask me.
With the digital upgrade to the Taken King, you will be entitled to play Destiny without the disc, but will still be required to download the complete copy.
Games run a little bit differently when you install them, compared to downloading them. Best thing to keep in mind: Games are made available online so you have easier access to them, not so that you can buy a cheap version of a retail game you own, and give the physical copy to a friend1. Games install from a disc so that the console has the files it regularly needs to access closer to home. This does not mean all of the files it needs.
When you install the game, it installs in a form that tells the console "when this game runs, check that this specific disc is in the drive, and use it as a reference point for other data". As far as the console sees, that install comes from a disc. It does not even check to see if you have a license to play the game online, as it assumes that you have the disc available to you.
While the online version would simply be adding to the original install, the two methods of playing Destiny are intentionally separate. For this reason, the online version does not "integrity check" the offline version, and vice versa. You could download the additional files from the download version of the game, except there is no way to check what files you have, and what files you do not.
The only real solution you have, here, is to simply download the entire original game from the PlayStation store2.
1Don't get me wrong, this is a perfectly good reason to do just that, but the developers have not set up their system to support this practice.
2My research to confirm the difference between an install and a download on the PlayStation 4 also reported that the PlayStation Store, itself, is often responsible for slow downloads. Given your reports that your internet is normally significantly better, I suspect the bandaid fix of "take the console to a friends house, who has better internet" would not be a good solution.
Best Answer
No.
While console transfer has been an option, in the past, this has only been used to entice users to make the jump to the next generation. Console transfer back to an older generation as not supported.
A Bungie representative has confirmed that this is how Destiny works, in answering a similar forum post.
In the same post, the Bungie representative links an FAQ, which has since been included to incorporate expansions, which should include The Taken King.
If you have not yet purchased Destiny: The Taken King: Legendary Edition, there is a limited time offer to purchase the PlayStation 3 version, and unlock the content for PlayStation 4. The FAQ also reports that you will use the same characters when playing within a console family.