Step 1: Create the small black hole.
You would need to focus 600 billion KG of energy into the volume of a proton. So, for starters, you will need "light" with a wavelength that is not much more than the width of a proton. This is quite a bit more than a hard engineering problem. It is not feasible - with any imaginable technology.
Step 2: Capture and redirect a significant amount of the resulting radiation.
That's the 160 petawatts of power - emitted in the form of anything - perhaps dark matter. If we get very, very lucky, there will be some way to bias the direction that this radiation is emitted so we don't have to work with the radiation except as reaction mass. More likely, it will be not feasible with any foreseeable technology.
Step 3: Make sure that tiny thing moves with the rest of your space craft.
I don't know how you would do this. It would be really tough to do it with gravitational attraction - emitting particles just barely forward of it so their trajectories bend back while pulling the BH forward. You would need a very high current of these particles aimed very precisely so they do not fall into the BH while getting close enough to make a meaningful tug. More likely, you could give the BH an electric charge and manipulate it that way. I might rate this one as potentially feasible.
If you ever did get this to work, you certainly would not let your BH evaporate in the 3.5 years described in the video. You would keep in well fed so that its mass would remain steady.
In this context a BH would be used as an exquisite fuel tank. A fuel tank to be useful you have to attach it to a vehicle safely and the in- and output must be controlled (yep, this includes the ability to 'shut it down' for some time when you don't need it).
Our physics right now has no idea how to achieve this: our engineering is even less capable. It is safe to say that this idea is just not feasible right now.