In that situation, I would try to force the spider out by spraying a little bit of insecticide through the 'Try Me' hole. If you don't spray too much, it should be enough force the spider out without killing it. However note that if it dies on it's web it will stay clung dead there so be careful.
Just note Spider behavior - first instinct may be to go to a corner and hide. It may then gradually re-assess it's situation (as it realizes it can't avoid the insecticide) and run out of the whole.
It is risky as it could die within the box but that may be preferable to having spider-webs and a live red back within.
Good luck!