I love this question so much that I can't stop thinking about it. Since this is a separate line of thought, I'll post it as a separate answer. Here's a general way to distinguish between DM stack whose coarse spaces are isomorphic with all the same residual gerbes.

Suppose a DM stack X contains a BG. Then the tangent space to the stack at that point is a representation of G. To get this representation, pull the sheaf of differentials of X back to BG. This is a coherent sheaf on BG, which is the same thing as a representation of G. Alternatively, there is a structure theorem for DM stacks (see Theorem 38.20 of my notes from Martin Olsson's course) that says that if G is the stabilizer of a geometric point x∈X, then there is an etale cover U→X of a neighborhood of x with a G action so that the neighborhood of x is the stack quotient [U/G]. I'm pretty sure that the action of G on the tangent space of the point over x is the same representation as in the other construction (or maybe it will be the dual representation).

Now the idea is that if two DM stacks have the same coarse space and the same residual gerbes, then you might still be able to tell them apart if they have different "residual tangent representations". In the three examples I gave in my other answer, the residual representations were (1) the sign representation of **Z**/2, (2) a two-dimensional representation (sign plus trivial) of **Z**/2, and (3) the trivial representation of **Z**/2. However, there is the annoying bit that you only recover G up to isomorphism, so if two representations differ by an automorphism of G, you can't conclude that the stacks are different.

Using this kind of thinking, we can construct two DM stacks which are both smooth, separated, and have any other kind of nice properties you can imagine, and have isomorphic coarse spaces with the same residual gerbes, but are non-isomorphic. Unfortunately, the coarse space is a surface.

Consider the action of **Z**/6 on **A**^{2} given by 1⋅(x,y)=(x,ζy), where ζ is a primative 6-th root of unity. This action is generated by pseudo-reflections, so by the Chevalley-Shephard-Todd theorem (the easy half of it), the coarse space of the quotient stack [**A**^{2}/(**Z**/6)] is **A**^{2}, with a residual B(**Z**/6) at the origin.

On the other hand, consider the action of **Z**/6 on **A**^{2} given by 1⋅(x,y)=(ζ^{2}x,ζ^{3}y). Again, the action is generated by pseudo-reflections, so the coarse space of the quotient stack is **A**^{2}, with a residual B(**Z**/6) at the origin.

But the two residual representations of **Z**/6 are non-isomorphic, even if you twist by automorphisms of **Z**/6. The first representation has a vector invariant under the action (namely (1,0)), but the second does not.