This is most certainly possible and technically feasible, if we slightly change our first assumption and ignore cost. Particularly if the race is highly cooperative and possesses great ingenuity!
I will address each of the concerns raised in the first two links mentioned above and of course the concerns raised here.
Lets start with building a massive, distributed nuclear fission thruster system; actually, lets not. Trying to move massive objects over great distances with propellant based system is a pipe dream. Instead, let's use our massive, distributed nuclear reactor to sustain our energy needs for millions of years. We will need it.
Since we still need to move the planet, what can we use? Well, we can use one of those shiny RF resonant cavity thrusters, or any kind of quantum vacuum plasma thruster.
On multiple parts of the planet, with the main thrusters being positioned within diametrically opposed bore holes (we've got to slow down too!). Only the opposed bore holes need to reach the core. The other thruster bore holes can settle for scaled down geothermal and nuclear power. We can use the smaller thrusters for thrust vectoring.
Using @iAdjunct 's 50 Gg of Uranium and the geothermal heat we have available will provide us plenty of electrical power to sustain our operations for at least a few millennia.
We will need to use @Thucydides suggestion of patience and interplanetary momentum transferring to get started. But once we've started moving, we will rely only on our thrusting mechanisms.
Since it will take some time to actually get moving (and for our engineers to invent these exotic thrusters), let's work on getting to the core. And while we're at it, let's consider some of the other challenges/opportunities we will face:
- Losing the atmosphere
We can start our work by bottling up the entire atmosphere. This will reduce the massive amount of pressure that would be on our thrusters & our underground systems. This partially removes the pressure problem raised in the second link above. Once we start moving, we will lose this atmosphere either way, so this is absolutely necessary for our long term success. And it will take some time.
- Surviving the conditional extremes
We need something to survive the extreme temperatures coming from below the thrusters, within the underground system (via the planet's mantle, core, etc) and from without and above (i.e the cold gas of a nebula, the heat from a near by star).
We need something to survive the extreme pressure as we build structures deeper into the planet, and as we travel closer or further away from massive objects.
Can we address all of these problems and the concerns raised by @D. Elliot Lamb and some of others in the second link? Turns out we can! We have our unobtainium: Aerogel composites!
Sufficient research could yield a host of aerogels with the necessary properties for most of our engineering needs:
Today's aerogel is already capable of maintaining functionality under extreme temperature differentials.
Aerogel's porosity gives it the mechanical properties necessary to bear high loads. Creating a composite with the right formula could allow our engineers to create a sort of planetary spring/sponge. This will be particularly useful for absorbing the forces created during acceleration and from tidal forces from other large planets.
As a bonus we can also store our atmosphere within the aerogel, killing two birds with one stone. Closer to the bore hole walls, the aerogel structures can absorb some of the steam produced from continuous wall cooling, using the aerogel as a thermoelectric catalyst to produce hydrogen and oxygen necessary for our chemistry and survival respectively (producing ammonia for food, breathing).
And while we're at it, let's grow things micro-organisms our aerogel pores :)
Eventually our scientists will figure out how to evolve a collection of organisms which can regrow what will become our symbiotic planetary host.
2a. The serious issue of building large structures
Even after removing the atmosphere, we still must support the pressure of our entire system as we move closer to the planet's outer core. Using Earth as an example, that's a 2,890 km (1,800 mi) overhead we have to deal with.
The composition of the mantle is locally a solid, but essentially a fluid over time. And its temperature (using Earth as an example) can range from between 500 to 900 °C (932 to 1,652 °F) at the upper boundary with the crust; to over 4,000 °C (7,230 °F) at the boundary with the core. [shameless copy pasted from Wiki]
Can we provide an anchoring solution strong enough to withstand the heat and pressure of geological time?
I think so, in the present moment we have (predicted) Hafnium Carbide superalloys with of melting points of about 7,460 degrees Fahrenheit:
And Hafnium superalloys are already known to be excellent an neutron moderators, to shield us from the neutrons produced from nuclear reactions in the planet.
Scientists are already pursuing Hafnium Carbide aerogels!
Having strong, porous materials along with an abundance of atmospheric gases can allow us to making floating platforms which withstand the large pressures from above. Technology like this also already exists:
In conclusion, by simultaneously constructing a planetary spring/storage system, a geothermal/nuclear energy system, and all the other planetary scale systems you'd need included, you could satisfy all of the requirements of creating a traveling planet which could sustain its own needs and survive the internal and external forces of traveling intergalactic space :)