I would put an extension pad of concrete or pavers so that instead of a straight sidewalk to make it a little wider and curve out to cover that small opening next to your gate. I think the dog likes to look out through that fence. She will always make a beeline there so there will be a lot of traffic and unless you have a grass that can handle heavy traffic, making a path along the natural pathway is probably a better choice in the long run than trying to keep grass there.
If you want to change the dog's path you can screen that part of the fence and change your gate to a chain link gate she can see through.
Seeded grass takes the longest to establish at about a year. Until the grass is well established and assuming you have a grass that can handle heavy traffic, it would be better to fence off the area to keep people and the dog off of it.
Happy gardening in Hawaii. Gardens are where people grow.