Uhm, the first thing that comes to my mind - regardless of whether this black hole propulsion might actually work - is the amount of lead we have available. Just building the spaceship seems to be just as much a problem as accelerating it..
Source: http://www.financialexpress.com/news/lead-uses-that-go-back-...
1) "More than half of the lead currently used comes from recycling." (from your article) - Once the spaceship is built and has left Earth, the lead is essentially gone (for the time being), so the more of those we build, the fewer lead we will have to recycle, and it will get even more difficult to build additional ones.
2) 400 tons for a small capsule, and consider they're proposing a spaceship in which multiple people could live in autonomously, i.e. they need places to live in, but also room to grow food, process their waste, etc. So I'd guess it's much higher than you seem to have assumed.
Oh and btw, I'm fairly certain we also need much of that lead on earth - hence why we're producing so much of it in the first place, so it's not like we have some spare lead in the order of magnitude of, say, 10000s (I'm really just guessing here, though) of tons lying around collecting dust.
The solution will probably come from materials science, but at the moment I don't think there's enough effort being concentrated on this problem to solve it in my lifetime.
Also, wouldn't being in such proximity to a black hole negate the need for time dilation effects of speed anyway? If you can build black holes, speed is meaningless as you essentially have a stasis machine. You could travel at .1c, reach Alpha Centauri in 40+ years and never even age a day if you can get close enough to the event horizon.
But hey, we're talking about highly hypothetical situations, so I'm sure we could make it work.
Of course you could also use water (obviously in much greater volume), which is a good thing to have in large quantities anyway while you're in space, and is evidently not all that uncommon, at least throughout our solar system.
If we made a large enough one say 1/2 earth mass, we'd get free "artificial" gravity on the spaceship.
The engineering details are left as an exercise for the reader :-)
"Dimensional Wave Super String Excitation Degenerated Radius Jump Gravitational Field Super-light Speed Navigation. WARP for short."
