Really the only way to control line width in exported raster images is to change the export size. The smaller the exported image, the thicker the lines will appear. You can adjust profile edge weight in Styles but that only deals with some of the edges. You might play with the Straight Line styles and see if they provide you with something use can use.
The alternative would be to spring for the Pro version so you can export vector lines and adjust their weight in some other application or go through LayOut.
Screen Size--1550 px Wide
800 px Wide
Straight Lines 0.2pix Exported at 1550 px wide
Straight Lines 0.4pix Eported at 1550 px Wide