If these are truly non-valuable items, you could go simple and use magnetic child locks. We did this on one of our pantry doors in our kitchen (as I did not want to drill into the front of them). We hide the magnetic “key” so that we and our cleaner knows where it is, and put a sign that says “owner storage, no guest access”. Could guest “break into it”? Yes, absolutely, and probably pretty easily, it’s just a plastic lock. But we haven’t had any issues so far. I also keep a set of keys hanging inside the door to our actual closet upstairs with a true lock and key, on the offhand our cleaner or our HVAC guy needs to enter up there.
I use the Jambini Magnetic Cabinet Locks, found on Amazon. What you have seems like it would work great too though.