So, an interesting thing about high eccentricity orbits like 2014 UN271 has, is that they have a much higher ratio of energy (kinetic and potential) to momentum around the sun than a nearly circular orbit like say, Earth has. One can think of an orbit as being a superposition of a purely circular orbit and a purely radial one, with all the momentum around the sun being in the circular one. Any leftover energy above what is needed for the circular component goes into the radial component. In this case, almost all the energy is in the radial component.

What this means is that if do you have a way to move it into a circular orbit around the sun, you could gain energy from the process, rather than having it cost energy. Maybe the best way to gather that energy, and possibly to move it as well, would be through gravitational assists, since they are lossless kinematic interactions. The energy gathered could then be used to build more of whatever is doing the interaction, exponentially speeding up the process.

Just for fun, I did some ballpark calculations. Let's consider the simpler problem of just putting 2014 UN271 into a circular orbit around the sun at its periapsis of 11 AU.

Calculating the exact path of an orbiting body under thrust is difficult. But the object will be within 1% of the desired orbital radius for roughly a 22-degree arc of its orbit centered on the periapsis, during which time its path will be closely approximated by a circular arc 648 million km long.

Let's say that over the course of this arc, we want to slow it from its initial speed of 12.7 km/s to the required circular orbital speed of 9.0 km/s. That means we need a continuous deceleration of roughly 0.00006 m/s^2 over a period of 2 years.

Assume that we'll produce this thrust by launching material from the object into space using mass drivers. By the Tsiolkovsky rocket equation, the smaller the fraction of the object that we want to use as reaction mass, the larger our "exhaust" velocity has to be. If we want to only lose 10% of the total starting mass, we need our exhaust velocity to be about 10x the total desired delta-V -- that is, 37 km/s, or roughly 0.01% the speed of light. This is a tall order, but let's say we can somehow solve the engineering problems and build a linear accelerator that can get rocks moving that fast.

Assume the object is 160 km in diameter and made entirely of ice, giving it a total mass of about 2.0e18 kg. The total required momentum change is therefore about 7.5e21 kg m/s, and the required energy input is 1.4e26 J. If we assume constant thrust for two years, this means we would have to launch about 3.2 million tons of material per second, averaging out to 2.2 exawatts of power required.

To put this number in perspective, it's several million times higher than the average electricity generation of the entire planet Earth. To generate this much power using 100%-efficient solar panels, at a distance of 11 AU from the sun, you would need a solar array approximately half the diameter of the sun itself.

So a direct propulsion approach, at least, doesn't really seem like it's within the realm of feasibility.

Yeah probably better off with a project Orion type approach.

We could aerobrake it around Saturn.

Scott Manley had a recent video talking about the work involved in moving the Earth, that you might find interesting. Basically the most "plausible" approach might be to move the moon and use it to drag the Earth. Similarly the best way to move another large object might be to carefully move other objects small amounts to do precise gravity assists.

https://youtu.be/5StYppuJ_lg

wouldn't that require a solution to the three body problem to make such precise gravity assists?

While we can't 'solve' the three body problem, we can make increasingly accurate estimations. Enough to usually get orbital mechanics right in the short-term.

For a setting where we were getting it right until we got it wrong, see Niven's "A World out of Time".

Only if the target was far and we did it in one shot. Otherwise you’d just measure the state of things and make corrections for your next maneuver, the way we do for trips outside Earth’s orbit today.

Nuclear would do it. Essentially use the dwarf planet itself as reaction mass, accelerating it with either nuclear thermal propulsion or a nuclear powered mass driver.

Highly impractical, and would take centuries to slow it down enough.

The Caplan Thruster of this video: https://youtu.be/v3y8AIEX_dU