I agree about NOT using Styrofoam, other plastics or rocks. Plastics can break down and let soluble chemicals leach out.
Thin twigs are great. I usually have some green stems that are stiff enough to help preserve air channels if I turn or even just "fork it" a bit.
I can also push a long-bladed trenching spade ("Sharpshooter spade"?) in all the way (around 16") and then lever the spade up and down to create gaps.
Since you keep saying "soggy", and that usually means very anaerobic, I would urge you to drape something loosely OVER the pile to make the rain run off. You don't want to wrap the heap in plastic - that would keep air out all by itself. Something more like a tent with open ends would be good.
If your rainy season will be over soon, that's another way to change it from a nasty fermenting mess to an aerobic COMPOSTING heap. Just be sure, once it gets dry enough to be CAPABLE of becoming aerobic, to poke holes or turn it once to break it up. And it has to stay damp enough in the dry season that "good" fungus species can thrive and EAT the bad, anerobic organisms that have taken over.
And, once it is draining and aerated, any water that flushes through the pile and EXITS will flush away most of the toxins.
That bin is as gorgeous as your beds! I think you made a great-looking garden even before you got the plants planted.
