First of all, congrats on choosing this route. I am sure you will love it.
I wore trail runners, as did the majority of others I saw during my August-Sept. walk. Results were mixed. There is a lot of road walking, and the shoes were hot during those times. I also didn't break them in enough, and the road heat and the improper socks I wore resulted in severe blisters on the ball of one foot. Wear wool socks, whatever shoe you choose.
One aspect of this route that doesn't get mentioned often, is how steep are the descents into town on road surfaces. If your foot is sliding forward, this will also create an opportunity for blister formation.
I'll be back on the Norte in 2025, and will stick with trail runners. Whatever you choose, make sure they are broken in, a half size larger, and paired with a good sock.
In spite of my blisters, some Compheed and a days rest, allowed me to continue along the trail. It is a life changing walk.
Good luck and enjoy.