The blinds should have been inset into the window recess. That usually means custom blinds.

What you request sounds like a Rube Goldberg solution to me.

Inside mount honeycomb shades would have been a darker cleaner fit. You have outside mount and this is the end result unfortunately. Never heard of the product you describe.

IMO add drapes to cover the gap and be done. I assume this is bedroom so I love the softness of drapes in a bedroom so just make sure the rod is wider than the blinds by at least 6' on eiter side get drapes that are 2 1/2 x the width of the rod and no more light gap.

Raise the shade and show the window. You may not have depth for an interior mount.

I'd follow @Patricia Colwell Consulting suggests...add drapery panels. Unlike the two double width panel shown here you can get away with one stationary panel each side, to block the light.

Here you go...

This is a youtube video about the product. It has side rails that prevent light leakage on the sides.