Much more than $1000 - Have you considered compensatory damages
Have you considered that there might be more to your situation than a $1000 vet bill? If your neighbors are entitled to compensatory damages the final bill might comprise their legal fees, your legal fees, and possibly other damages for emotional distress, the hassle of driving the dog to the vet, etc.
In that instance, you could easily incur a judgment for many thousands of dollars and much stress. There are attorneys out there that literally do nothing but "dog bite" cases and might jump at the chance to represent your neighbors in a case where you're clearly at fault.
If you're trying to avoid a lawsuit you'd almost certainly lose, why make such a meager initial offering? Are you really too cash-straped to pay more than $5-$10/week?
If this were my problem, I'd profusely apologize and find a way to pick up the $1000 vet bill immediately, as opposed to hoping my neighbors aren't inspired to sue by my pathetic $5/week settlement offer.
Please note, the above writing is based on my own individual opinions/research and is not professional or legal advice of any kind. Follow any implied suggestions at your own risk.