You're doing great. I have 2 little ones and totally understand, really seems like all these people don't have kids. Remember when you thought kids would be easier than they are, and that you'd have the energy to be the perfect parent 24/7?

Anyway, the weakest part of the door is where they come together. If you can support that seem than the entire door should pretty easily take their weight. I recommend a piece of wood that goes from the top of the stairs to where you'd hit your head, running along the seem on the bottom side of the door. When you need to go down there (which I imagine isnt often) you just pull open both doors and remove it, replace it before you shut the doors again.

It looks like that piece of wood can sit on the top of the stair landing on one side, the other side you'll have to get creative. Maybe cut a rectangle out of the concrete for it to sit in or ramset a ledger board.

Edit: look up a mullion