OK, here is my suggestion even though I have to wonder why a fence is built with such huge gaps unless the terrrain is hilly.
Go to your local lumber yard and obtain some lattice. It comes in pressure treated, Cedar, and composite. Determine what would work the best for you. The woods can be stained to match the fence. Here is a page of them with various designs etc.
Here on page 2 is lattice that is 2 feet by 8 feet long.
Then you can cut what you need to fill the gaps. You can even bury the lower portions of the lattice in the ground. Even though the pressure treated and cedar are resistant to rot, they both will eventually deteriorate. But it should out last your dogs.