I'll try to work out a trajectory as close to the source material as possible and send you the scenarios. Unfortunately, Orbiter time during the weekdays is very limited, but I should have them ready by the end of the weekend.
Not sure if it's actually possible to follow the show's trajectory. It has them intersecting Pluto near perihelion. (It shows distance from Earth as 4.47 billion km, which means even if Pluto and Earth are in a heliocentric conjuction at that point, Pluto is a maximum of 4.62 billion km from the sun, which is just outside Neptune's orbit. It won't be that close to the Sun again until about 2219 (and then again after perihelion in 2249.) Also, Pluto's argument of perihelion is about 90°, which means when it's near perihelion, it is significantly above the plane of the ecliptic, not near it as the trajectory graph in the video suggests.
Also, all those transfers look close to brachistochrone curves, so this ship must have massive delta-V. It can pretty much go anywhere and do anything. As an example, my back of the napkin calcs show that their trans-Venus injection would have taken about 32 km/s.