A couple ideas:
Use cold-ish water if you are not doing so already. You might try dropping your water amount to 62% as you have another 1% effective hydration from the oil. Try making this dough exactly as you list it above, skip the poolish step, and mix everything at once. If it is still sticky, try a second batch with a higher protein flour such as bread flour (I do not know what you are currently using). If that does not improve things, I think we need to determine which strength ldmp you are using. If you have 20L 1% should be ok. If, like me, you have 60L then you may need to reduce the amount you are using by two-thirds.
One or more of the changes above should get you something less sticky. That’s on the front end of your dough making: immediately after mixing. To combat “way too airy,” I suspect you should get your dough balls into the refrigerator right away rather than after 4 hrs at RT. For 500 g flour I would go to balls right away rather than bulk. 22 or more hours later pull it from the fridge about 2 hrs before you want to use it and let it warm, as DoughDoctor says, at room temperature. I think this will give you a dough that has a consistently more even crumb with fewer voids and large bubbles.
That is quite a few small changes as ideas. Try one thing at a time and keep track of how things change. There are people here a lot better at this than I am. Hopefully we can keep the conversation going and hope one of them sees this. I don’t think you are far off, honestly.
EDIT: meant to say after paragraph 1: once you get a dough that is less sticky and handles better we can deconstruct your formula and work in using a poolish again. It will be easy. Good luck.