I think I've got the most silly idea and yet so wonderfull and genius to avoid any cosmetic screw-ups whenever having to glue this issue. You need a straw that you can squeeze it enough to be flat, any tool to put the cotton/polyester back to it's place, stick the straw in, cut it off just enough and then pour the glue in.
I made pictures to explain it better. With this way, you'll avoid it messing it up.
You need any very flat tool to put the cotton back inside:
https://i.imgur.com/whfsfxp.jpg
https://i.imgur.com/8CWN0Eo.jpg
A straw that you can cut whenever you feel like it and make the end flat:
https://i.imgur.com/bHZpHIX.jpg
Cut it somewhere half but you can decide it yourself, this is mine:
https://i.imgur.com/WbRWzOZ.jpg
When you're satisfied on how you placed the cut straw, you can pour the glue in the straw and squeeze or move whenever you feel like it and make it dry a bit. Make sure not to move a lot. It depends on which glue you use but this is mine:
https://i.imgur.com/yGoYsmc.jpg
This is my first time I tried it this way so there is a slight visible place but fortunately the headband is very dark to spot it easily. Next time when fixing the same issue on other headphones other than Amirons, you know what to do.
After finishing it:
https://i.imgur.com/4vWbVta.jpg
https://i.imgur.com/XPK0hzm.jpg
After it's dried up, you can move the headband cushion to hide the spot.