I am using AC24 7000AUS full. I followed the steps above (new file, one grid line, one wall, one dimension (auto-dim to core faces) from wall to grid). I saved the whole file as a mod and placed it into a new file. It worked fine.
On my work files, the module system is a bit different. My first floor is moduled and placed above. Rather than working with a bunch of separate files, I saved the L01 as a view that has a specific module dedicated layer combination. I got a publisher set for mods, so whenever I operate changes on my L01 and want those changes to update across the levels I publish that view/module and update the hotlinks. It works perfectly but for dimensions. Also, I ensured that the grid layer is turned on in my module view (anything ON is moduled).
Could the issue come from the fact that the module origin and instances are in the same file and somehow Archicad won't module the grid? I had a look in 3D and it seems that everything is placed above but the grid which would explain why the dims are broken.
It is suspicious that the grids are not duplicated above. In a new file (same Archicad version and build as you are using) I was able to save the ground floor as module then place it on the first floor while having the grid duplicated. I also had my dimensions show up correctly. I even tried to save the dimensions and the wall only (the grid was not included in this new module) and put it on the floor above: it had all the content of the module. I think this issue might need deeper investigation. I suggest reaching out to your local Graphisoft representative here for further support. Please send them the problematic file, this detailed description and maybe a video showing the issue in detail for better understanding.