For the geometry you're showing, aiming for something non-destructive but simple, I would make the side walls with a rim-only solidify, and then weld the two copies of the sidewall that would be created by that solidify:
Showing 0 vertices non-manifold after convert to mesh.
For a small trade-off of non-destructive vs simple, you can drive the thickness of the solidify modifier from the distance between vertex-parented empties.
For different geometry, with normals that weren't at right angles to the main plane at the margins, this technique would still be possible, but more complex:
We do a miniscule solidify (because normals depend on non-zero-area-faces) with assignment to a VG, then modify our rim normals to point properly so we can do a displace from custom normals, weld (which will set some of our rim verts to weight 0.5) then finally mask away the unmerged rim verts.
The normals are measured from empty to origin, so the empty should be at the origin except for the Z axis, and should be parented to the object.