Nate, I doubt that the dough is deflating b/c you are wrestling it out of the container, but if that is the case, then you might want to oil your container a bit better including the walls, invert the container and let it sit 20-30 seconds and the dough should fall out.
Two other causes of dough deflation I can think of is 1) the gluten is develop sufficiently and unable to hold air or 2) the dough is a bit overfermented and there is breakdown of the existing gluten structure causing the dough to deflate.
In your case, it could be #2 especially if you are using a starter and the end result is a (overly) sour crust. If your crust isn't overly sour, then I suspect it could be a combination of #1 and #2 at work. +4% oil is a bit high for a NY dough IMO. Oil also hinders gluten developement to some extent.
I would try 2% oil, keep your mixing the same. Once the dough comes out of the mixer, cover it and let it rest 10-15m. Pull the dough mass out and add 4-5 stretch and folds to the dough. Basically you are gently balling up the entire dough mass here. Now bulk it as usual. When you go to divide the mass into balls, you may also choose to ball the entire mass again "gently". Meaning don't make a tight ball, but just enough stretch and fold to give it a loose or semi ball shape. Let it rest 20-30min before dividing and makind individual balls. It's okay that after you make the balls that they spread out again in the containers. That is normal.
Basically everytime you are balling or reballing after a rest period, you are building more strength into the dough. If the dough is slack as it is, then it may need more strength. You can do that several ways. Drop the hydration, increase the protein content of the flour, decrease the oil, increase the kneading, don't ferment so long if you are using a natural starter (or switch back to IDY or ADY).
In your case, I think it may be a combination of your starter and a lack of gluten strength. BTW, after balling up the individual dough balls, I typically let it rise about double max if using a starter. 150% rise maybe too much rise for a natural starter. The acids that accumulate can dissolve the dough and toughen the crumb texture.