What about taping a piece of cardboard between frame and the door jam? This would prevent the door from fully "closing". Of course, this does mean that leaving the screen door closed with the glass door open will have a notably "unscreened" portion.

Seems easy to replace

Are you actually up on a higher floor where there's little chance of anyone trying to get in that way? If the lock has a key, maybe you could hide a copy on the balcony? Alternatively I would wedge something into the lock, it's not like it actually DOES anything. Anyone who wants in your house will simply go through the screen without making enough noise to alert the neighbors.

Are you able to remove the hook on the door frame that the lock engages with?

Brainstorming here:

• fasten a magnet to it, then use a magnet on the outside to slide it up as a "key".

• remove the locking mechanism

• undo the screws on the mechanism in the screen door and add some sticky residue like a piece of duct tape then remove the duct tape by using a hair dryer to get it off. Then reassemble and put it up in the unlocked position and let the sticky hold it in place

A nice dab of hot glue holding the lock up would work. Or some superglue

Maybe take the handle off and put some thin fabric/felt/velcro fuzz behind it to give it a little more friction?

Looks like it relies just on friction to keep it in place and it's worn enough to not work 100% now. I agree with others, a screen door isn't a security barrier, so unless it needs to be secured to stay shut for wind or whatever, I'd just permanently stick it open. Glue, or perhaps taking it apart and then jamming something between to add more pressure.

Wedge something under the lock so it stays up, then put tape over it to hold it in place.