What's the best method to model this atomizer with varying size and thickness?

Question

I used the tissue modifier to make this but the mesh needs to be thicker and bigger at the end points, like this:

Answer 1

The hosiery on the bulb is thinner on the swollen area because it is stretched. One approach to simulating this would be start off with a base-tube, and stretch it ourselves:

This can be proportionally edited as a shape-key, or just on a duplicate.

Once made, the stretched version can be compared to the base version with a GN group, storing the ratio of lengths into a vertex-group:

(In the end, here, the attribute has to interpolated onto points, not edges, which isn't perfect. But that's why it's OK to stash onto a pre-existing vertex-group, which can be directly used by the Tissue tesselation)

A weight-map on the bulb results:

When you tesselate using the Tissue add-on, you can ask for the vertex-group to be mapped to the tesselation. This method tesselates a tileable mesh-line, without thickness:

Then uses a GN modifier on the tesselation to generate thickness, guided by the mapped vertex-group. Ideally, as a band is stretched, to preserve volume, the cross-sectional area varies as 1 / length, and its radius varies as the square root of that:

The result is subtle, but you could fudge the square-root to another Power to exaggerate it.

This is the sort of result:

I'm not that fond of the workflow.. it has destructive steps. You could do both the stretching and the tessellation inside GN to improve it, but that's another question.