Why is my white bread with poolish dough very loose?
The protein content of your flour may have something to do with it, but it may also be the fermentation of your poolish. The more fermentation a dough goes through, the more the yeast turn starches into acid. Acids can denature gluten and gluten-forming proteins, leading to a less elastic dough. When you're adding your poolish to the rest of the ingredients, you're also adding its acid. The poolish is always going to be goopy and loose, so switching to bread flour for the poolish may not have the effect you desire. 

Unfortunately, I'm more or less in the same boat. Like you mentioned, a higher protein content may help. That said (if I remember correctly), Forkish recommends AP flour for all of his recipes. Perhaps a shorter poolish ferment, or shorter bulk ferment.