Please create a new layer and set the desired Lineweight for this layer:
Call the Construction note (or any other) and click Global options:
Set the newly created layer as the default layer for these Notes:
That's all:
You can also change the layer to the desired one for all Notes at once in the same Global options window. Or open the PARAMS window and do the same on the Symbols -> Notes tab.
For Multileaders, Lineweight is configured in the manager:
To use these parameters in other drawings, save the drawing with the settings you need as a DWT file and set it in the program settings (Options window) as default: