Civilly, your liability exposure is limited to repairing "the property" (as dogs are viewed in most jurisdictions), which means vet bills. If the victim is special in some way that makes it more valuable, and that value is lost or reduced (show dog, valued breeder), then you could face that exposure, as well.
This should be a handleable amount and you should consider leaving your homeowners insurer out of it, if you can. They'll certainly pay it, but may drop you (try getting a new insurer if you've been dropped) or may condition your coverage going forward on a dog limitation of some sort (including "not that dog", not a GSD or other "aggressive" breed, not a dog over 20 lbs, etc.). The Very Real downside to not telling your insurer is that not telling them breaches a precondition to coverage. If the neighbor is really mad and files a suit (even one that might ultimately not succeed), you'd have to hire a lawyer or risk a default judgment (bad). If you then told your insurer, you could be told "sorry, you didn't report it promptly, so no coverage due to lack of compliance with conditions, best of luck." You'd then pay for the lawyer on your own. Or, your insurer could go ahead and pay for this claim, but drop you (see, "bad to be dropped," above).
Non-civilly, the neighbor could involve the authorities and what happens to your dog will depend on the laws of your jurisdiction. You may be confident your dog wouldn't do to a child what he did to the other dog, but you likely will not succeed in convincing your neighbors. In any event, neighbors don't like living in fear of a dog and they may attempt to make life rough for you and your dog until the dog's gone.
Tough situation, to be sure. Be sympathetic to the frightened/angry neighbor, offer to pay any/all expenses (may include neighbor's time off work, travel, etc. to give the dog care, vet trips, etc.), maybe offer to muzzle the dog when it's out of the house, that kind of thing. Anything to appease him/her and stop additional response.
As for the dog - I'm no expert, but it sounds like obedience training and more socialization may not impact either dog aggression or high prey drive. Under these facts, I'd consider finding someone who handles drivey dog issues like this for some one-on-one sessions, setups for failure w/E-Collar correction, and the like. S/he may suggest modifications to the home environment that permit the dog to believe it ranks at the pack's peak, thus making the world your dog's oyster.