I think the length of the diagonal of the unit square cannot be in the same class as transcendentals such as pi. I can make the length of the diagonal equal to exactly 2, for example, if I make the length of its sides equal to sqrt(2).
Further proof: A circle should be viewed as polygon with infinite number of sides. I can reduce those numbers of sides to 4 for a square and look at the diagonal like a proto-"diameter", then the square's diagonal cannot be a irrational, because this proto-diameter of the square cannot bear the same irrational and transcendental relationship as the diameter to a circle without violating the geometrical perfection of a circle.
What should be said about it then? The diagonal would be put into a new category of unknowable, rational numbers. It is unknowable because the relationship between diagonal and the length of its sides must retain indeterminate regardless of scale, yet non-transcendental because of the relationship noted above in reference to circles.