Edit: This answer plays around with one extreme of the question with a large orbital radius and a large orbital velocity that immediately self-destructs. It probably misuses special and general relativity. Also, it requires a civilization with energies at it's disposal beyond the Kardashev scale. Please take it with a grain of salt. It is a thought experiment gong wrong. (Though even negative thought experiment results are still results.)
Let's suppose you give the planet a large orbital radius R. As R approaches infinity the effects of the black hole approach zero. What you're left with is a planet moving through the vacuum of space. Now the question becomes "how consistently can a planet moving linearly through space be used as a clock". Nobody knows how many meteors are moving through the void between galaxies so let's get back to it later. Now the only things that can change the velocity of the planet are radioactive decay from the planet's surface, fluctuations in radiation pressure from the stars coming from the night sky of space, quantum tunneling, and unpredictable variations caused in the gravitational warping of space time. Most of these effects are extraordinarily tiny.
Radioactive decay from the planet's surface can be lowered to below humanity's ability to measure by using non-reactive substances. Quantum mechanics becomes unmeasurable once you hit macroscopic scales, on planet-sized scales it is completely irrelevant. Radiation pressure is stronger than both of these and the random pressure from solid objects hitting our planet even if it's just cosmic dust is greater than radiation pressure because an individual dust particle imparts much more momentum than an individual photon. The momentum of a cosmic dust particle is about 10km/s * 10^-4kg = 10kg m/s. The momentum of the greatest cosmic ray ever measured has an energy of 3 x 10^20 eV. Divide this by the speed of light and we get momentum of 1.6x10^-7kg m/s, a difference of eight orders of magnitude. The effects of cosmic dust are smaller than the effects of cosmic meteors. How many cosmic meteors are out there? Nobody knows.
But the cosmic dust and cosmic rays don't really matter because you can minimize their effects by speeding the planet up to near-lightspeed. The faster you get the planet going the less small changes in momentum are going to affect it's velocity, thanks to relativity. (Though you do eventually end up producing a nuclear explosion with energies far beyond known physics. At this point you have a bomb, not a clock, and what you're measuring is the shockwave of weirdness.) So if you get a planet moving at 99.999...% the speed of light you can make the clock arbitrarily precise, except for variations in it's course caused by unpredictable gravitational fluctuations warping spacetime. These come from dark matter and dark energy, stuff we also don't understand.
So the answer to your question is that by bending the rules of this situation beyond the breaking point we can make the clock arbitrarily precise (at least up to the Plank limit beyond which time and distance cease to mean anything). The precise accuracy of the clock for a given set of circumstances is far beyond known physics.
But what if we didn't stretch the rules to the breaking point and instead used a simple situation that we understand very well? What if we took an Earth-like planet and placed it around a sun-like star with similar Earth-like conditions? Then the orbit would vary about as much as Earth's does.
Edit: Neighboring Galaxies
Joe asked if the orbital radius would be large enough to be affected by neighboring galaxies. I didn't think of that in my original answer. The answer is yes, so it looks like we'll need a Kardashev Type IV civilization to push around some galaxies to give this clock some space.
First. Let's calculate the orbital radius. We'll start with GMm/R^2=F=mv^2/R. (I'm not sure if this equation doesn't work for scales this large. In fact, I'm not even really sure what'll happen to orbits at relativistic speeds.) This equation reduces to GM/v^2=R. The mass of the black hole in the center of our galaxy is 8.2×10^36 kg. Let's plug this into the equation and use the speed of light as v. This gets us R=5x10^21m. The third-closest galaxy to the Milky Way is 1.6x10^21m so this orbital radius on the order of the distance between galaxies in our neighborhood.
However, the distance between galaxies varies and we're not in the most empty part of the universe. Cosmic voids contain few to no galaxies and have a diameter or 10 to 100 megaparsecs. R = 5x10^21 meters = 0.2 Megaparsecs (Mpc). So you could fix the lightspeed orbital clock in one of these with a 5 Mpc radius to spare. This means the neighboring galaxies may be 20 times farther away thereby exerting at least a 400th of the gravitational force of the black hole at the center of this clock. That is significant.
Technically what we need to worry about here isn't gravity from neighboring galaxies but rather gravity from dark matter which there is more of and that will produce a stronger force. We don't know how the gravitational pull of dark matter fluctuates because we can only measure it's long-term effects on large scales that places dark matter beyond the realm of current science, so let's just focus on the matter galaxies for the moment.
They will certainly throw the clock off significantly if the civilization building this clock but will they do so unpredictibly over tens of billions of years? The universe is only 14 billion years old. I doubt we can predict galactic movements that far out, but you'll have to ask a Kardashev Type physicist to be sure.