I have Destiny, including the Taken King, downloaded on PS3. What is the least I must buy to play it on PS4

You did not get ripped of. You got what you paid for.

From what I could tell, I found that in the bundled items it displays:

• PlayStation®4 System (Glacier White)
• DualShock®4 Wireless Controller (Glacier White)
• Destiny Video Game (Physical Disc)
• 30-Day PlayStation Plus Trial
• HDMI Cable
• Power Cable
• Wired Mono Headset
• USB Charging Cable

I could not find any mention of the "Digital Guardian Edition" so they are not entitled to give you the additional expansion pack.

You may have received the disc for the expansion pack version, and the armory code (although expired), you did not buy the Digital Guardian Edition so your not entitled to anything extra.

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.}