I think what Harry says is heartfelt, but it's also a decent false trail to prevent Dumbledore from accidentally working out the secret and losing his ability to cast a Patronus.

Not that Dumbledore necessarily needs that. He's in a great position for doublethink: he can presumably use the Pensieve, label the memory "the secret of dementors and the Patronus charm," then Obliviate himself. Locking the basilisk away in a secret chamber, if you will.