Is the above $\mathbf {B}$ a solution to the GS equation?

The answer is no. The above magnetic field is not a solution to the GS equation even if the plasma pressure in the GS equation is set to zero. The above expression is a solution to the GS equation in the limit of infinite aspect ratio and zero plasma pressure. Finite aspect ratio and plasma pressue introduce the Shafranov shift (see Sec. 10).