It is so hard as a parent, when a child is labeled cortically blind, to figure out if or what your baby is actually seeing. This is an interesting web page that shows some examples of what children with damage to the optic nerve may see. It is not necessarily what our children see, but helpful in visualizing what their vision may be like. Surgery, drugs, and therapy may help some of the conditions. In terms of specialists, ophthalmologists are trained in surgery and eye diseases, so their opinion would lean more toward surgery.
Optometrists tend towards therapy.
Surgery being rather drastic, therapy should be considered first. Surgery would help exotropia and endotropia only, not the other conditions. I was not able to find much about what drugs might be used or in what way they would work. One interesting idea from a research study is the use of Botox for esotropia and exotropia. Regardless of what it is called the idea is that vision is a learned skill and can be rehabilitated to a certain degree.