Unfortunately if you want to stop the bindweed you're going to have put something along that border that goes really quite deep into the ground. Whatever you put in the gaps, the bindweed will find its way through or under. Is there anyway you could talk to your neighbour and perhaps share the cost of getting a professional (who will have access to commercial grade chemicals) to spray it off for you?
As for the gaps themselves, didn't the contractor who erected the fence discuss this with you before starting work? In sloping gardens fencing panels aren't normally recommended for this very reason. Not helpful to you now I know but something for future reference.
You could consider doubling the concrete gravel boards under the fence - you will need to dig one end into the earth but it would last a long time.