Can I achieve good cycling shoe stiffness in winter with a boot insert?
It's not a very good idea, but since you're going to do it anyway, I'll give you some pointers on what to expect.


The contact area between your foot and the surface of the boot / pedal system needs to somehow conform to your foot, or you will have painful hot spots. In your idea, the sole of the running shoe should take care of that, and the insert can be indeed flat. The outsoles of running shoes are curved, though, so your stiffener would have to conform to that shape or you'd need a lot of room inside your overboots.
The shock absorption in running shoes absorbs a bit of energy on each pedal stroke, feels mushy and doesn't really offer any benefits for riding. A plate between running shoe and outer shoe wouldn't help here. This is why cycling shoes usually have thin anatomically shaped soles.


My suggestion would be mountaineering boots or hiking boots with rigid soles combined with large platform pedals. These boots are very stiff, don't have too much shock absorption and are designed for low temperatures.

As a side note, thin plastic plate isn't going to be very rigid. The stiffness of a plate is related to 3rd power its thickness, so in order to have stiff soles you'll need either a very thick plate or 3D structure. Because of the exponential relationship, using stronger materials doesn't help much.