Two sided reinforced "clamp" for creating a hole in screen
Yes, look up &quot;cat door for sliding screen door&quot;. These have a 2 flat frame pieces that snap together, suspended within the screen mesh itself. They also come with a lightweight swinging screen door that hangs from the frame.
If the cat door is too large for you, you could go down to the hardware store and have them make a tiny window screen frame. You would then use the frame spline channel and rubber spline to suspend the frame into your screen wall, cutting the screen fabric from the middle to make a hole. I would actually cut the sides and bottom of the fabric and let the flap hang to keep bugs out, maybe weighting it at the bottom.