Quick & dirty door latch idea
Get a dowel/closet rod, match the door thickness in the bottom corner to the "Jamb" thickness, drill a hole in the door or jamb to match the dowel diameter, make the dowl 1/16th longer than the door is thick, bolt it together with a long bolt. If made as drawn, the weight of the levers should keep it latched.
HOT TIP When drilling the dowl, drill from one end half way in, then meet in the middle from the other side to minimize "off centerness" (a technical term for the hole not being in the middle) If you try to go all the way through on 1 shot you will end up with a way off hole on the exit end, if you meet on the middle, you will have 2 centered holes on each end and a crooked tunnel which can easily be reamed out oversize to make your hole sized long bolt fit through. I used to build stair cases a lot and had to drill "Through holes" in post bottoms and handrails all the time.
Also, when drilling the dowel, For the love of cripes....put the dern thing in a vice or clamp it down very firmly, because when you break through the tunnel, it will invariably jerk like heck and possibly stall your drill, that kind of torque will mess you up. My 1/2 inch Milwaukee Hole Shooter has tried to injure me on numerous occasions, I handle that thing with a ton of caution.
If you put this on one or both corners, it can be opened from inside & out and locked from the inside if you wish with some simple pin thing er sumthin.
Hope this makes sense. Should only cost you a buck or 2