Repairing your original button is your only option here. Although yes, it is true that only Apple can reprogram a home button, it doesn't help because they don't work on iPads. Yes, that's correct, Apple does not fix iPads; best they'll do is offer you a trade-in or discount on a replacement unit.
And no, there is no aftermarket service that can do that reprogramming; it requires a proprietary encryption algorithm that only Apple has access to.