I agree with the idea of putting the reason right back on the kids:
"Sorry, neighbor, I talked it over with my DH but we're just going to have to say no. OlderGirl and DS aren't a good combination - they just feed off each other, and DS ends up being frustrated and angry. I'm definitely not saying it's all OlderGirl's fault, but it does mean they spend the whole time screaming and bouncing off the walls and even with DH and me both here, it's too much chaos to handle. Add in the younger kids and it's a recipe for disaster! Here's the number of a local sitting service I've used before, though - they're really good about finding someone even on late notice, and we were really happy with them."
By giving your neighbor the number to a sitter/sitting service, you're changing the request from "Will you help us out in a pinch?" to "Will you LET US BE CHEAPSKATES and work for free when we have a perfectly viable option already set up in a pinch?" It makes asking you a much harder request for your neighbor to make, and a much easier one for you to turn down ("Sorry - did I give you that sitting service's number, though?").