Your dough really shouldn't be fighting you on the stretch. There's two areas where you can boost the dough's extensibility and produce something that's easier to form. Gluten development and hydration.
Cold fermentation (refrigeration) is a kneading equivalent. Forgive me if this has been discussed before, but if you're cold fermenting for two days, you need very little kneading at the offset. If you're using the bread machine to knead the dough until it's smooth, it's way too far. Smooth is great for same day doughs, but when you add refrigeration to the mix, it's kneading overkill. You want to let the machine knead the dough until it's well mixed and no more. Cottage cheese-y to somewhere between cottage cheese-y and smooth is a good appearance to shoot for. My total mixing/kneading time, by hand, is about 2 minutes. I don't know how this translates to a bread machine, but I would dial it back- maybe try half as long and then checking on the appearance of the dough.
Also, just to confirm, you're forming the final dough balls before refrigeration and not re-balling the dough anytime close to forming, correct?
You're working with KABF, right? KABF can handle a couple more percentage points of water. Go with 62% hydration. That should help with extensibility.
Other than that, everything looks pretty good, if you're current amount of yeast is almost doubling the dough after two days cold ferment, that's right on the money. Water encourages yeast activity, so, if you bump it up to 62%, keep an eye on it, but I think you should be fine.
Here's my tips for stretching dough:
Watch this video a few times (ignore the rolling pin stuff)
When you're flattening the dough ball with your finger tips, leave a slight mound in the middle.
Gently slap this small mound a few times without pressing the dough too much, this will activate the gluten in the center of the dough, and when you pick it up to knuckle stretch, the middle will stay tighter and won't have a tendency to thin out as much.