Have you never been in a situation where you are about to take train going at 80 mph; wondering when you'll meet up with your friends going in the opposite direction at 90 mph?
I believe also the launch vehicle has a launch window (the moon moves 15 degrees per hour) and launch trajectory. I'm lazy so I'd compute the extends of the launch vehicle's motion in the sky (from earliest possible launch to latest possible launch), and then intersect that geometry with the moon position geometry without explictly trying to solve the equations simultaneously. That should back-project to shapes on the ground at which point you could reasonably expect to be able to get a good shot, and then you'd do some adjustment in your pointing in real time.
A smart college senior could do it directly (IE, not lazily compute a bunch of points and manually intersect them on a screen).
I didn't know, for the longest time, what a derivative was. I still don't, not really, though I can bluff my way to a reasonably correct answer using the above analogy and going from dry "rate of rate of change" sort of language into more practical concepts.