Your mechanIsm for making the Moon collide with Earth in 100 years would not work.
LONG ANSWER PART ONE: ANOTHER SOLAR SYSTEM OBJECT COLLIDES WITH EARTH
As other answers have said, you can not hit the Moon with a single impact that will change the orbit of the Moon so that the Moon will crash into Earth in 100 years.
But it is perfectly possible for astronomers in the present time or some future time to discover that an astronomical body is going to impact the Earth at some specified time in their future.
Some present day astronomers are working on projects to discover and catalog as many asteroids and comets with Earth crossing orbits as possible. And they are calculating the orbits of those asteroids and comets as precisely as they can and re observing them to get more data to recalculate their orbits better. The hope is that if some asteroid or comet is calculated to be on a course that will eventually impact on Earth with devastating results, the calculated impact date will be far enough in the future for the asteroid or comet to be diverted in time.
For example, Biela's Comet had an orbit that crossed the orbit of Earth:
The comet appeared as predicted during its 1832 apparition, when it was first recovered by John Herschel on 24 September.1 The orbital elements and ephemeris calculated by Olbers for this return created something of a popular sensation, as they showed that the comet's coma would likely pass through the Earth's orbit during a close approach on October 29. Subsequent predictions, in the media of the time, of the Earth's likely destruction overlooked the fact that the Earth itself would not reach this point until November 30, a month later, as pointed out by François Arago in an article designed to allay public fears.2 Despite this, the fact that Biela's Comet was the only comet known to intersect the Earth's orbit was to make it of particular interest, both to astronomers and the public, during the 19th century.
In the 1820s it was calculated that Biela's Comet would collide with Earth in the year AD 4339.
The Year 4338: Petersburg Letters (Russian: 4338-й год: Петербургские письма) is an 1835 novel by Vladimir Odoevsky. It is a futuristic novel, set in the year 4338, a year before Biela's Comet was to collide with the Earth as computed in the 1820s although the comet burned up later in the nineteenth century. This work was originally conceived as the third part of a trilogy, which was also to have featured depictions of Russia in the time of Peter the Great and in the author's contemporary period, the 1830s. The first part was never written and the second and futuristic parts remained unfinished. Fragments were published in 1835 and 1840, with the fullest version appearing in 1926.
Biela's Comet split in 1845 and the fragments were last seen in 1852.
And on the other hand, in recent decades, a few of the Earth-crossing asteroids that have been discovered have been discovered only after they passed very close (by astronomical standards) to Earth and were moving farther away. Thus it is certainly possible that an asteroid or comet headed for Earth might not be discovered before the impact.
So in a fictional story where an asteroid or comet is discovered to have an orbit that will result in an impact on Earth, the characters might possibly have hours, days, weeks, months, years, decades, centuries, or millennia to prepare for the impact.
A writer can choose an arbitrary length of time for humans to try to divert an incoming asteroid or comet or to evacuate the Earth if the impact can't be averted. Of course it would be better if the writer got someone with expert knowledge to calculate an orbit for the object that makes it reasonable for it to be discovered his chosen length of time before the impact.
And of course the writer should make an effort to chose a size and mass for the incoming object that will make it impossible for any reasonable technology to divert the object in time to prevent an impact.
I believe there was recently an question about how big an incoming object would have to be to make the impact unavoidable by anything humans could do. Here it is:
What order of magnitude incoming asteroid could we, in fact, deflect?2
Can we move the moon to use as a shield?4
I note that there are objects classified as "centaurs" orbiting beyond Jupiter, that seem to have characteristics of both asteroids and comets. There orbits are usually rather unstable due to perturbations by the giant planets, so is it is easy to imagine that if some extrasolar dwarf planet passed through the other solar system it might perturb a number of centaurs into orbits that pass into the inner solar system, including Earth-crossing orbits.
And some of the centaurs are very large, tens and even sometimes hundreds of miles in diameter, so a large centaur calculated to impact with Earth would be very difficult to divert.
I note that billions of comets are believed to orbit the Sun in the Oort Cloud far beyond the planets. Occasionally comets are diverted into the inner solar system by the gravity of nearby stars, etc, and any comet that enters the inner solar system for the first time might possibly have an orbit that will make it collie with Earth.
Comet nuclei can have dimensions of less than one kilometer or mile up to several miles. In extreme cases a comet can have a radius of 30 kilometers (19 miles).
So it is certainly possible for a comet from the Oort Cloud on a collision course with Earth to have thousands of times the mass of a normal sized comet and thus be thousands of times more difficult to deflect.
Furthermore, some of the centaurs might have originated in the Oort Cloud, since they have similarities to comets. And the largest known centaur, 10199 Chariko, probably has a diameter of about 250 kilometers, which might be about 50 to 100 times the diameter of a typical size comet, making it have about 125,000 to 1,000,000 times the mass of a typical comet.
If there are Chariko-sized objects in the Oort cloud, and one of them has been perturbed into a collision course with Earth, it would be many thousand times more difficult to divert than a typical sized comet.
Anyway, it is the job of good writer to make the readers understand what it would take to divert an object from a collision course with Earth, and when the object in the story is too massive to be diverted, make the readers understand how hopeless it would be to try diverting it.
LONG ANSWER PART TWO: A ROGUE PLANET COLLIDES WITH EARTH
Remember that it is always possible for a writer to create a larger incoming astronomical body than can be diverted by any human effort. Our Sun and solar system orbits around the center of the galaxy like many other stars do. And just like it is possible for two solar system objects to collide as the both orbit the Sun, it is possible for two different solar systems to collide as they both orbit the center of the galaxy.
Of course astronomers have calculated the courses of all the nearby stars and know how fast the distances between them and the Sun are increasing and decreasing, and they know that none of them is going to collide with the Sun in the next ten million years.
And surveys of the sky in infrared light have detected some very dim brown dwarfs close to the solar system, and it now seems very unlikely that any brown dwarfs close enough to become a danger in the next few million years would still be undiscovered.
For decades, science fiction writers have imaged there might be "rogue planets" that don't orbit any stars but travel alone though interstellar space. In When Worlds Collide (1932,1933) by Philip Wylie and Edwin Balmer, a rogue planet from interstellar space collides with and totally destroys Earth, so that plot has been used in fiction for at least 87 years.
Astronomers have actually detected rogue planets that don't orbit any star orbiting the center of the galaxy by themselves.
Astrophysicist Takahiro Sumi of Osaka University in Japan and colleagues, who form the Microlensing Observations in Astrophysics and the Optical Gravitational Lensing Experiment collaborations, published their study of microlensing in 2011. They observed 50 million stars in the Milky Way using the 1.8-meter MOA-II telescope at New Zealand's Mount John Observatory and the 1.3-meter University of Warsaw telescope at Chile's Las Campanas Observatory. They found 474 incidents of microlensing, ten of which were brief enough to be planets of around Jupiter's size with no associated star in the immediate vicinity. The researchers estimated from their observations that there are nearly two Jupiter-mass rogue planets for every star in the Milky Way. Other estimates suggest a much larger number, up to 100,000 times more rogue planets than stars in the Milky Way. A 2017 study by Przemek Mróz of Warsaw University Observatory and colleagues, with six times larger statistics than the 2011 study, indicates an upper limit on Jupiter-mass free-floating or wide-orbit planets of 0.25 planets per main-sequence star in the Milky Way.
Nearby rogue planet candidates include WISE 0855−0714 at a distance of 7.27±0.13 light-years.
Rogue planets could form in interstellar space similarly to how stars form, or else be ejected from solar systems by gravitational interactions. And of course meteoroids, asteroids, comets, moons, dwarf planets, etc., can also be ejected from solar systems into interstellar space just as well, and much more often, than planets are.
Suppose that an interstellar object is detected approaching the solar system and Earth at a relative speed of 1 kilometer per second, and is detected 100 Julian calendar years before calculated impact with Earth.
The object will travel 60 kilometers in each minute, 3,600 kilometers in each hour,
86,400 kilometers in each day, and 31,557,600 kilometers in each Julian calendar year. So it will travel 3,155,760,000 kilometers, or about 21.0949 Astronomical Units (AU), in exactly 100 Julian calendar years.
That distance would be between the orbits of Uranus and Neptune, though an extra solar object would probably not be approaching from the plane of the ecliptic.
If the rouge planet orbits the galaxy in the same direction as the Sun, it would probably have a relative speed compared to the Sun of less than about 100 kilometers per second, I guess.
Thus the rogue planet could travel about 315,576,000,000 kilometers, or 2,109.4952 AU, in exactly 100 Julian calendar years between being discovered and collision with Earth.
Of course if the rogue planet was orbiting the galaxy in the opposite direction to the Sun, it might have a relative velocity of several times 100 kilometers per second and might possibly be detected at several times a distance of 2,109.4952 AU.
A writer would want his approaching rouge planet to be small enough not to have been detected yet, but large enough to have been detected with the lead time he wants for the story - in your case 100 years.
Astronomers have searched for additional planets in our solar system and the lack of success so far has establish limits on on large and how close such hypothetical planets could be.
As of 2016 the following observations severely constrain the mass and distance of any possible additional Solar System planet:
An analysis of mid-infrared observations with the WISE telescope have ruled out the possibility of a Saturn-sized object (95 Earth masses) out to 10,000 AU, and a Jupiter-sized or larger object out to 26,000 AU.7 WISE has continued to take more data since then, and NASA has invited the public to help search this data for evidence of planets beyond these limits, via the Backyard Worlds: Planet 9 citizen science project.
Using modern data on the anomalous precession of the perihelia of Saturn, Earth, and Mars, Lorenzo Iorio concluded that any unknown planet with a mass of 0.7 times that of Earth must be farther than 350–400 AU; one with a mass of 2 times that of Earth, farther than 496–570 AU; and finally one with a mass of 15 times that of Earth, farther than 970–1,111 AU. Moreover, Iorio stated that the modern ephemerides of the Solar System outer planets has provided even tighter constraints: no celestial body with a mass of 15 times that of Earth can exist closer than 1,100–1,300 AU. However, work by another group of astronomers using a more comprehensive model of the Solar System found that Iorio's conclusion was only partially correct. Their analysis of Cassini data on Saturn's orbital residuals found that observations were inconsistent with a planetary body with the orbit and mass similar to those of Batygin and Brown's Planet Nine having a true anomaly of −130° to −110° or −65° to 85°. Furthermore, the analysis found that Saturn's orbit is slightly better explained if such a body is located at a true anomaly of 117.8°+11°
−10°. At this location, it would be approximately 630 AU from the Sun.
Many of these limitations also apply to a possible rogue planet approaching from interstellar space.
But astronomers have mostly searched for planets orbiting in the ecliptic plane or relatively close to it. A rogue planet from interstellar space could approach from any angle, including angles far from the ecliptic plane.
Thus it may be possible for your rogue planet to be closer when detected than a planet in our solar system might be while remaining undetected.