Yes, you can play Destiny TTK at the same time, because when you download a digital game onto a different console, it adds a "temp" license for that user, as long as the account that has bought Destiny is on the console the game should be fine.
I hope this helped. Good luck Guardian
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
So the Taken King is by downloaded/accessible through a code that you enter in to the PS Store. Chances are if you got this from a public library the first person who took the game out was the lucky dog who got the code for free, or the library took the code out. Codes are a one time use, so even if it was in there and the code did not work you would be in the same boat. You will only have access to what the disk gives you.