It definitely is fast growing. I had a plot with the same circumstances. I nuked it and started over. One thing you do NOT want to do is allow that stuff to set seeds. You'll never win the battle then.
2-4D is also an option. You'd kill everything but your rye, but your rye is likely going to be thin with everything else going down. I'd look at doing 2-4D first. I'm getting nervous about using glyphosate consistently.
3rd option is to just manage it as a pigweed plot. Keep mowing it before it sets seed. Deer will eat it. I don't know enough about pigweed to know when it'll set seed though.
Then it is probably Glyphosate resistant for sure. 2,4-D will be the cheapest route to go if you spray 12" and down. 2 pints and 20 gal water per acre. But you may have up to a month of residual that come along with using 2,4-D. Liberty herbocide would be a great choice at about $15-20 acre, and you do not have to worry about residual carryover. Good Luck!
I was under the impression you could mow and mow to burn out the seed bank lying on the surface, but such doesn't seem to be the case according to Iowa State. If it's never been plowed before, perhaps a pass with the mow board plow could get ya ahead of it.
I can't speak for our property up north because we don't really have any of it (knock on wood). Back at my dad's in McLeod county, what little there is gets browsed. Them boys play for keeps when it comes to spraying around there. We can however grow about 50 bu/ac french weed with remarkable consistency.
A guy can try anything he wants to get rid of it. But the day will come as many have found out, nothing will work to control it. Just be thankful if you can suppress it!
I would recommend changing your spray strategy in the very near future or you may end up with resistant pigweed as well. Many types of pigweed have resistance to some herbicide or another, proving they are a highly adaptable species.
I'd hit it with Trimec (or similar), then throw down some more winter rye in a month or so. Down there I'd think that would be plenty of time to get it established.