This won't allow to slide the card around, it'll just stick to screen in one place. Better learn how to get the screen off entirely, because this will happen again
Yeah I was thinking the best thing to do it just unscrew it, hold it upside-down and shake it out. You have a point on how would you rig it so it does not happen again.
I agree with calling the guy to get it out, but a close second would be to put thin piece double-sided tape around the tip of a bamboo skewer feed it down in press the card to it and pull it out. if one skewer isn't enough, tape a second one to the first one
Get a full sized piece of paper that you can slide in from the top, that shout teach the card but can still be pulled out from the top. Use tape to stick the card to the paper. Then pull it out.
If NirodhaAvidya's vacuum approach doesn't work out for you and you can't find an easy way to get the screen door apart, a couple sewing needles (or something similar) might do the trick. Stab the card toward a corner, rotate it so the far corner is higher up, stab with another needle, remove the first one and repeat to walk it up toward an opening where you can grab it. Might have to pick a new spot to poke it every few iterations if you're not getting enough friction to make the card move.