If it was only about the diameter at the base, this would solve it:
Edges can be filtered and the direction force can be used to force them to be parallel to the Z axis like this:
rajno_diameter_vertical.gh (43.3 KB)
Adjusting the height is more complex. From looking at your structure, I don’t think 12 meter maximum height and 10 meters arch height between the bases is going to make much sense as the center part does not have much curvature which to me seems weak.
Quad Remesh might actually not be the best way to create the base mesh and a semi-manual process might be required to have more control over the direction of the edges.
Here’s an alternative solution with a base mesh I created with a radial grid and some manual tweaks in Rhino. It’s also not perfect but in contrast to QuadRemesh you have full control over singularities. Those are the points where in this case more or less than four quads meet.
rajno_diameter_vertical_radial.gh (71.0 KB)