How to lock this closet with a non-standard knob size?
I can't tell what the existing backset (distance from front-side edge of door to center of knob shaft) is, but you can buy mortise locksets that lock. A bit of chisel work would expand the existing mortise.
I shudder to offer this link, but as an example of widely available things, mortise lockset
The linked one appears to have a 2-1/2&quot; backset.
I'd suggest an architectural scrap yard, except that their locksets don't always come with keys...