Andrew, although I admire your ingenuity, I doubt very much that your plan would work. There are multiple reasons why:
1. Brown Button/Crimini Mushrooms/Portabello (Agaricus bisporus) do not grow from stem butts onto cardboard, at least according to
Paul Stamets. Their mycelium is very weak and is not aggressive.
2. Button mushrooms are among the most difficult of all mushrooms for amateurs to grow. They have a very narrow temperature and humidity range for growth and fruiting that is difficult to duplicate without spending a lot on a special building or chamber to grow them in.
3. Button mushrooms require a high quality horse or cow manure
compost that is well-pasteurized and free of any contaminants. This is difficult for beginners to make. As far as I know, they simply won't grow on just chicken manure or cardboard. (However, chicken manure is sometimes added as a supplement to the high quality horse or cow manure compost that is used in Button Mushroom production.)
In my own
experience, I prefer to grow Almond Agaricus (A. subrufescens) instead. These mushrooms are much, much easier to grow, are much better for you and in my opinion, they are one of best-tasting mushrooms I've ever tried, with a flavor that is far better than the Button Mushrooms at the store. The Almond Agaricus can also grow from stem butts onto cardboard and it is much more forgiving when it comes to substrates. Chicken manure could be added to the cow manure compost that these mushrooms prefer, too. I grow Almond Agaricus inside my house in cardboard boxes filled with leached cow manure compost I get from a
dairy farm. I buy the spawn from Field and Forest Products in Wisconsin. There is a
thread in the fungi section where I explain how to do this. The only bummer is that the Almond Agaricus does not like our chilly Pacific Northwest nighttime temperatures, so they must be grown in a
greenhouse or warm room. They can be grown outdoors in the summer in the Southeast and in coastal California, even the Midwest. I hope this info helps and doesn't disappoint you!
Here is the link to the thread I just mentioned:
https://permies.com/t/33807/fungi/Growing-Portabellows-wild