Keeping the padlock quiet

I was curious if anyone knows how to silence a padlock. I recently acquired a new cage that mirrors my old one, but it uses a separate padlock rather than an integrated lock. The built-in mechanism used to create a pinch point and leave me with sore spots. Since I usually wear boxers at home, the audible click from the current setup is bothersome when children are nearby. It’s fine if it’s just my wife and me, but not ideal around kids. Does anyone have a way to secure the padlock quietly? I considered swapping it for a zip tie, but that wouldn’t give the same locked feel as an actual padlock.