I've thought about this before - do you actually need to "shoot it down" (make it explode)? What if you just nudge it a little and either make it spin or change its orbit? If your missile can reach the satellite then these seem like things that should be possible, no?

Depends, if you nudge it only a little, its own onboard stabilizers / thrusters should be able to correct it. It'd have to be more than its own systems can correct for.