Putting this together with the net force, you get the momentum principle. It’s kind of a big deal in physics.
So, it seems like you could solve this problem for the motion of the mini moon. Just calculate the force, use that to find the change in momentum and then use that momentum (and the velocity) to find the new location of the mini moon. Yes, this would work—but it’s actually impossible to obtain single equation for the position of the mini moon. The difficult part is that the mini moon pulls on BOTH the Earth AND the moon. This means that their momentums also change. All three objects interact with each other and it’s just not solvable unless you make some approximations (like deciding that the force on the Earth is too small to worry about).
This problem is actually so famous that it has a name. It’s called the Three Body Problem—and we can solve it. I know what you are thinking. I just said you can’t solve, right? No. I said you couldn’t get an equation of motion for the three objects. However, I CAN find the position of the stuff at certain times. The way to find out how these things move is with a numerical calculation. In numerical calculation, the problem is broken into a whole bunch of short time intervals. During each time interval, we can assume the gravitational forces are constant (even though they aren’t). With constant forces, it’s fairly easy to find out where the objects are at the end of the time interval. Then, moving into the next time interval, I can just find the new force (since all the objects moved) and assume it’s constant again.
This might seem like you are getting a solution at no extra work, but this method comes with a cost. If you break the motion into 1 second time intervals and you want to find out where the stuff is after 100 seconds then you would need to do all these calculations 100 times. So, instead of one impossible problem of finding the equation of motion, you get many simple problems. But at least it’s possible.
Personally, I don’t want to do endless calculations for the motion of these three objects. However, I don’t mind making my computer do it for me. In fact, no one does these kind of calculations by hand anymore. Many people might even call it a computational physics solution. I think it’s important to keep the “numerical calculation” name so that no one thinks you HAVE to use a computer—you don’t.