oh goodness. I would have the wire replaced with glass . I dont see how any wire screen could remain lint free near a laundry room.

Have you tried a shop vac with a soft brush attachment? I think its too late for a can of compressed air if you've already tried other wet resolutions.

For window screens, DH takes them down, outside. He washes gently with brush and soapy water, rinses using one of my garden hose attachments that has a soft spray. It amazing how much pollen and dust is drawn into window screens, even without the clothes dryer adding lint to the scene.

If you did take your door down, I'd wipe it dry immediately with old towels. Meaning the door itself if it got wet, I wouldn't be too worried about the screening.

That appears to be regular screening, not chicken wire, but, I would think a brush used gently, with some dishwashing detergent, should clean it. It cleans window screens, and even the grate on my Hvac filter, nicely.