How many hours of direct sun does that location receive daily? Between the houses, trees and vinyl fence it looks too shady for veggies.

I don't see anything wrong with chain link there, but check if you have neighborhood rules about matchy matchy fences.

Leafy veg and root veg might be the only thing there's enough sun for but you probably know your microclimate better than us, I'm spoiled by all my sun into thinking that's the only ideal conditions.

Do a chain link fence and plant vines on it to cover it up. If this is the front of the house, then do a wood gate with chain link fence covered in a vine. Hopefully there is an evergreen vine that grows in your area. Here's my chainlink fence on the right, covered in a vine. I dd wood across the front. If you really want to go budget wise and this is the back yard, do a chicken wire fence with posts covered in vines with a wood gate.

Did you intend to place a second fence inside the white vinyl one? Not sure I understand the reasoning or the need for that. Just a partial fence and/or gate across the open portions is all that may be required. And a fence is not going have any impact on a digging dog :-) Unless it has a portion that is buried below ground.