Like other views (Base/Projected etc.), Flat pattern view does not have 'Shading' option in the drafting. The following can be a workaround. There may be some other way to achieve the same,not sure about it.
First, Turn off the 'Monochrome' display of drawing.
Start the 'Area Fill' command. Set the selection mode to 'Boundary Curves' and select the flat pattern view boundary. Under settings, select pattern as 'Solid Fill' and the desired color.
How to color a flat-pattern ? or make it Fully Shaded ? (In drawing)
I'm curious to the value of this, can you explain? If it is to make the outer profile clearer, you can user a thicker line width for the Exterior line definition.