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
I believe the only way to confirm a game is fully installed is to put the disc in and wait for it to install fully. By which I mean, put the game disc in (but don't start it) and you'll hear the disc start to spin, if it's fully installed, after roughly three minutes you'll hear the disc stop spinning, if it doesn't stop, it's still installing and will stop the disc when it's finished.
Similarly when you first put the disc in it will keep installing from the disc and when it's done will stop the disc which is your cue to know it's done.
That's the only way I've ever been able to tell, the install progress bar is useless because that just tracks the install time of the data required to run the game, not the whole install. The size on disc is equally unhelpful in this regard as each game will reserve space for its full install from the off, even if it's only currently installed a fraction of that.
All that said I've not noticed any particular issues with games only being partially installed and I can't see that extending beyond the first couple of sessions.