So it's just a matter of keeping the chickens out of the dogs' space, not actually keeping the dogs out? That makes sense--I can see how a chicken-proof fence would be enough for that purpose, without needing to be dog-proof.
Might be easier and cheaper to fence the garden than the chickens?
Yup, that probably calls for fencing the chickens, although it also depends on how close the neighbors are.
You might be able to let the chickens out shortly before dark, and then they won't have time to wander very far before...
It looks like the no-shock fence is meant to control where the chickens can wander (keep them out of your garden, or off your porch.) But when I looked it up, the page says it will not stop predators.
So it all depends on what you want to fence to do--contain the chickens vs. protect them too.