So one small hack later and I have it working.
At line 152 of wp-content\plugins\flickr-gallery\flightbox\jquery.flightbox.js, change
$("body").
append('<div id="floverlay"></div>').
append('<div id="flightbox"><div id="flightbox-right"><div id="flightbox-next"></div></div><div id="flightbox-left"><div id="flightbox-prev"></div></div></div>');
to
$("body").
append('<div id="flickr-gallery">');
$("#flickr-gallery").
append('<div id="floverlay"></div>').
append('<div id="flightbox"><div id="flightbox-right"><div id="flightbox-next"></div></div><div id="flightbox-left"><div id="flightbox-prev"></div></div></div>');
You can then style the overlay and main lightbox container in your theme's css by something like
#flickr-gallery #floverlay {
opacity: 0.8;
}