The question is how to scale an existing mesh without changing the thickness of the walls?
I am using Blender to create STL files for 3D printing. Let's say I create a shell for a model railroad car. Since 1/87th is the most common scale I make the walls of the shell just thick enough to make it rigid in 1/87 scale. Now, if I want to print the same shell in a larger scale, say 1/48, the wall thickness will nearly double and it will waste material printing walls that are thicker than needed. If I want to print in 1/160 the printing may fail because the wall thickness falls below the minimum the printer will support.
Your question falls into two different categories, here at 3D Printing SE and there, at Blender SE.
I would consider that your objective would best be solved using some form of parametric modeling, an aspect that is rarely embraced by Blender. Even though the limitations of Blender make life interesting for you, there may be a couple of useful features within (and without) the program.
On Blender SE, a question of similar format exists, with a somewhat open-ended answer. A quick search using The Google, with the terms "Parametric Modeling with Blender" results in a number of different approaches. According to a quick perusal of the search results, some of the solutions involve free plug-ins or add-ons for Blender. More complexity rather than less, perhaps.
I'm familiar enough with the very simple basics of Blender to know I would not be able to make use of those answers. I'm also well aware that Blender's power extends beyond my own limitations with features supporting scripting, animation and so many other tools. Seeing the workflow diagrams/charts that make up some of the advanced portion of the program leads me to believe that one can accomplish your objective, but one must be a certified wizard with the program.
As an alternative, one could engage any one of the many parametric modeling programs available. I'm a fan of OpenSCAD, although the text/scripting interface can be daunting for some. If you've become skilled in Blender, a non-GUI format isn't necessarily the best route, although the GUI options are no less confusing, in my opinion.