You want to use gravity (Mr. Woo's weight) to close the door.
"Box" traps use varied triggers and doors. Rabbit sized ,with a smaller sizes trigger maybe in your case.
And what about fishing line replacing your string? Or just longer string.
A simple Pigeon trap involves a hole they been using to go in and out already in a wall or box or frame. A bird keeper could use such to keep them in if desired.
Hanging on that hole/opening (think of strung beads as a curtain) are stout wires (coat hanger wire for us paupers).
A length of supporting wire across the top. (Yes Bob Cratchit's coat hanger wire would work.) Consider it a curtain rods purpose , sort of...
Eyes bent round in one end of each wire to be hung will be supported by that cross wire. Keep bending until you have enough wires to cover the opening. (remember the illustration my therapist presented me? Alright , try to forget it.)
Wild guess , 1/4 inch (6.5mm) gaps between the wires will suffice.. (See illustration for potential cut soda straw spacers location between wires at top.)
Important:
Each hanging wires bottom edge must hang below threshold of the opening on the INSIDE of the compartment/box/cage ect. Making them swing only one direction. Bird can easily push through following it's visibility of your bait or it's simply returning to it's former shelter.
Trying to exit though it pushes wire bottoms against the bottom edge as designed.
Should work on your cage. ?
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Then there are the stick and string capture boxes of cartoon fame. Yes they work.
As kids we'd use a torch (battery powered flashlight) to catch roosting birds in the dark.
Use the light only to locate roosted bird if possible. Holding light on a bird wakes it more, but also gives it plenty of light to do what birds do when you annoy them in daylight.
Meaning you climbed the outside of the silo for nothing. Again.
Then with minimum (light only on edge of periphery if your eyes won't adjust to dark) put the sneak on the bird.
Grasp by pinning wings to body securely but gently.. Easy does it.