When showing that the magnetic field derived from the curl being proportional to current and the divergence being zero is unique, Purcell says that at some sufficiently remote enclosing boundary the difference of two magnetic fields (two possible fields from the curl), D, would take a constant value. Why is this true?
I know that this doesn't directly answer your question about Purcell's reasoning (see addendum I wrote after reading Purcell's argument), but here's how a uniqueness proof would go.
Suppose that two fields $\mathbf B_1$ and $\mathbf B_2$ both satisfy the magnetostatics equations $$ \nabla\times\mathbf B_i = \mu_0\mathbf J, \qquad \nabla\cdot\mathbf B_i = 0, \qquad i = 1,2 $$ Let $$ \mathbf D= \mathbf B_2 - \mathbf B_1 $$ then the curl and divergence conditions give $$ \nabla\times \mathbf D= 0, \qquad \nabla\cdot\mathbf D = 0 $$ Now your question reduces to what we can say about the vector field $\mathbf D$. Take the curl of both sides of the first equation and use the following vector calculus identity: $$ (\nabla\times(\nabla\times \mathbf D))_i = \partial_i(\nabla\cdot \mathbf D) - \nabla^2D_i $$ along with the zero divergence condition, to obtain $$ \nabla^2 D_i = 0 $$ In other words, each component $D_i$ satisfies Laplace's equation. Now, if we assume that the magnetic field components vanish at infinity (as would be the case for a bounded current distribution) then we recall that the only solution to Laplace's equation that vanishes at infinity is the zero solution. This gives $\mathbf D = 0$ and thus $\mathbf B_1 = \mathbf B_2$.
Addendum. After having read Purcell's argument, I'm fairly confident that he also is making the physical assumption that bounded current distributions produce fields that vanish at infinity. Such a boundary condition cannot be derived from the equations themselves.
Purcell is rather vague when he specifies this boundary condition; what he says before the argument is "We don't consider sources that are infinitely remote and infinitely strong."