There is no reason you cannot have conventional double doors on a closet.
Bracing the Inside Corner
To solve the free corner issue, you can put a small stop, attached to the floor at the center point between the two doors. It needs to be wide enought to stop both doors (probably anything over 1 inch would do) and tall enough to catch the doors at the bottom (3/4 or 1 inch should do, and you want it as short as practical).
The simplist approach is to use ordinary domed door stops screwed into the floor inside the closet. One could probably be placed to catch both doors. These may be prone to a toe stub, but are probably the least likely to cause harm.
![domed stop](https://i.stack.imgur.com/900Su.jpg)
If you want to make your own stop, you could use a metal L channel or a wooden block screwed through the top into the floor. Metal is smaller, but may be more prone to hurt if you stub your toe against it. A small wooden block with sloped shoulders might be better. It only needs to be thick enough to hold up (maybe 3/4 inches). All exposed edges sould be slightly rounded with sandpaper to soften the edges for safety.
Latching
The simplist approach to holding the doors closed is to use magnetic latches. You can put them at the top only or at the top and bottom. If you do put them on the bottom, put the latch on the door and the strike plate on the floor stopper (another reason to use a wooden stopper).
![mag latch](https://i.stack.imgur.com/bHzbP.jpg)
As an alternative, you could use a cabinet latch at the top.
![cabinet latch](https://i.stack.imgur.com/g99Dc.jpg)
Finally, you could use a button type latch set into the tops of each door and the door frame. These are somewhat harder to install, but look more finished.
![button latch](https://i.stack.imgur.com/Iz5Wa.jpg)
Handles
In each of these cases, you would use dummy handles or pulls to open the doors.
I think you've hit on the easiest solution. Putting in a layer of insulation board and then boarding over the top of that is the approach I'd take.
If you don't want to use the door then you can just put insulation right to the frame and board over that to hide the door completely. However, if you want to be able to use the door then you'd have to leave enough space at the edges so that the door could open without catching on the frame.
Insulation board is quite light so you'll probably be able to glue a frame to the door an inch or so in from the sides and then put the insulation in that. A piece of hardboard glued or tacked over the top will keep the board in place. This will mean that there is still some heat leakage around the sides but nowhere near as much as through the entire door.
Best Answer
Polyurethane Foam. The type that is used for insulation.
This works because the foam will keep the hollow walls from collapsing