New beam in old exterior wall

[freppe.ar]

I have an odd question about how I should proceed. I need to insert a stud in an exterior wall (I suppose) to be able to screw up new drywall.

The interior walls are clad with particle board, but in the corner against the exterior wall, there is a missing stud, something to screw the new drywall into.

Should I cut the plastic, remove some insulation, insert a stud, screw it in as best as possible (only accessible from one side) at the top and bottom, and then tape the plastic together on the new stud?

Any potential dangers with this?
Anyone have tips on an easier approach?
Things to consider?

 

[Maestrono]

The ruler is supposed to be recessed behind the wall to the right. Try putting your hand behind and feel around.

 

[freppe.ar]

The stud is likely recessed behind the right wall. Try reaching your hand behind and feel around.

Absolutely, there's a stud if I reach my hand in, but it touches a bit up on my forearm, so unfortunately not one I can screw into.