As far as I know you cannot make the extent indicator background transparent.
For a workaround, I'd suggest picking a marker fill symbol background. In my example, I used a 1pt circle marker with 2pt of separation (it's a mega-zoomed inset map). You could use other symbols for more bleed-through but this is what I've used in the past when transparency is not an option. Line fills look okay too depending what your underlying data looks like.
