Any cloth bag with a loose weave will also work. Closely woven bags are a second good choice, but won't breathe as well.
Part of the reason for a basket, though, is that you'll also have a lot of structural support, and the goods will have less crushing, bumping, and bruising. You can use plastic fruit containers that have built in drainage and air holes ...
Most of the mushrooms you will find in a guide are low risk because they don't have similarities to poisonous ones - if you have a good guide it should point out the ones where identification is more difficult.
Avoid those ones and you will be much safer!
I have good experiences using cloth grocery bags (note: breathability).
Depending on how many mushrooms you are gathering it might be necessary to put something like a bowl/small basket into the bottom of the bag to keep the mushrooms from getting squished by the downwards pull of the bag.
I am quite certain that there are more deadly mushrooms than just the one you mentioned - if only for the other members of the Amanita Genus, many of which are very poisonous and some plain kill you.
E.g. Amanita Vitrosa, with the uncommonly poetic name destroying angel (German: Weisser Knollenblätterpilz), is just as deadly as the one you mentioned. What ...
Also known as Bolbitius vitellinus, that there looks to be a young Yellow Field Cap. They're quite small when they're yellow, but they quickly mature into taller, flatter, brownish mushroom.
Those look like oyster mushrooms. They are the right size and color and have the right type of gills.
Pleurotus ostreatus, is a common edible known for its oyster-shaped cap. One of the first things you should look for when trying to identify this mushroom is the presence of decurrent gills.
Decurrent means that the gills are attached to and run ...
You could actually try a mushroom bag, which is basically a cloth bag with a plastic liner that usually has a draw string to keep it tight. My mother in law has one, she uses it for shop brought mushrooms sometimes (you can also get similar potato ones) but the idea then is you can store them straight away.
They tend to stop sweating and moulding of the ...
I propose an onion bag or any other sack with a loose mesh fabric. Along with allowing air to circulate around the mushroom, it allows any spores the mushroom may release to return to the wild. By allowing the spores to escape, you're allowing the mushroom to reproduce and be there for the next harvest.
Hard to tell from photo, but: "gills" look yellow which is wrong for a button /agaricus compestri . Do they have gills ? If they have a spongy matter and not gills they are likely some boletin . Usually found under conifers. If so, likely edible but not much flavor.