I had leaky windows in an apartment and since I wasn't going to be opening them all winter, I put a sheet over them (plastic doubtless would be better) and taped it around the outside of the woodwork. I didn't put any tape on the front or main surface of the trim, only 'around the corner' on the narrow edge that gets paint slopped on it when you paint the walls. It would be so much easier to draw this, but I can't. Trim is 3 or 4 or 5 inches wide by 1/2 or 1 inch thick, right, so the tape went on the half-inch part. Have I managed to describe that at all?
The sheet used to inflate slightly with the air coming in....
If you have plastic on the outside, and put another layer inside, you'll have made a little airspace between the outdoor air and the indoor air. That will be super insulation, although possibly a moisture problem . . . I would try fabric on the indoor side, or leave a small opening along the taped edge of the plastic for it to breathe. The moisture will gradually equalize, but you'll still be insulated.
Yeah, you might think it would occur to people to do something besides sit around saying "Gee, there's a bunch of cold air coming in here." I'm lazy, but I also hate being cold. I hope the foam board lining works on your window seat. I'm rooting for ya!