I also know how unlikely this situation is, so please refrain. From telling me "it won't happen."
Jupiter is sent on a spiral into the inner Solar System. Both Earth and Mars enter orbit around the gas giant and extreme chaos is caused now by the unexpected tidal pull of Jupiter and the other moons. Jupiter settles into a stable elliptical orbit around the sun a little closer than our current position. We were amazingly lucky.
Now, how lucky are the nice people in the moon colonies? Is there a better chance that our moon became one of Jupiters' moons, switching from orbiting the Earth to orbiting Jupiter? Or is it more likely that the moon and the unfortunate scientists on it were swallowed up by Jupiter? What would be the timescale involved assuming the Moon stays within Earths orbit for a short period before being ejected from the Jupiter System? Assuming the moon people have technology fifteen years more advanced than modern tech, what is the maximum "point of no return" or the point at which they have no choice but to evacuate?
I have decided that the Earth will enter the orbit of Jupiter at 32,000 mph and maintain that orbital speed. The moon should be traveling at 4,800 mph roughly. I'm not sure if this helps or not.
I would prefer the moon entering a stable orbit for at least a short period of time (maybe a few months to a few years) and then what happens happens.