Joseph,
That's a big question there--will the Wine Caps do anything to the chemical nasties that you suspect are in the straw.
I just did a mini deep-dive and the answer is probably not. That part is sad.
BUT
There is still hope if you want to remediate contaminated straw (but not use for garden bedding just yet)!
It turns out that Oyster mushrooms utilize a different set of enzymes and they are likely to break down something like 30%(ish) of the chem. nasties that you might find in the straw--especially those stubborn persistent ones. I would not use this for a garden, at least not yet, but if you wanted to get a jump start on breaking this stuff down in what will have to be a multi-year process, my first step would be to go ahead and utilize the Oysters to break down the straw. That straw is then going to have to sit in time out for a while longer.
After that first year, the next step is to give all the natural decomposers the best possible advantage at breaking those chemical bonds. So we throw everything at them that they don't like. That includes:
Sunlight
Moisture
Soil microbes
So the technique that I found that might be the best looks something like this:
1) Go to some part of the land that is at least somewhat fertile--you want the microbes that make that soil fertile. If you can, clear the ground right to the soil. Don't worry about eliminating grass or existing vegetation completely, but get to the ground so you can spread out the composted straw. Maybe slightly disturb the ground (avoid tilling, but scratch a little) in order to give extra surface area. See if you can collect maybe a quart of good topsoil from that area. If you can't get it from that area, get some good topsoil with mijcrobial activity from somewhere similar.
2) After spreading thin, mix in that quart of topsoil--just giving the soil microbes every opportunity to thrive.
3) Consider planting a sacrificial cover crop for one season--the microbial action in the roots will boost the activity breaking down the nasty chems.
4) If you are really feeling ambitious, here are some bonus steps:
A) Add compost tea--MICROBES!
B) Bokashi
C) Add some almost-boiled rice mixed with a little milk--you have a solid material that has lots of simple sugars for microbes to grow and thrive.
By the end of year two you should have really broken down the chems. You probably still want to have this checked, but it should be ready sooner than without all this help.
Maybe worth a try?
Eric