I think the problem are the different toolbars for different devices/screens. If the sibling item is gone on smaller devices, also the chev button is gone.
The exact problem is that the editor changes the toolbar, like it literally re-parses all the bar when doing mobile so PUP losses its position and binding.
Is there a way to go back to the older version of PUP where you could place the button outside the box?
I got it figured out and it was definitely looking great but, now, since the update the button is no longer showing.
EDIT: For now I manually swapped the PUP for the 3.15.1 version. It seems to work.
if it's placed on top of the post reply and preview buttons it would also not ruin mobile responsive themes, that are deformed since the chevereto PUP button makes the width of the page too large. Just a suggestion. Or if there was a way for the PUP button to somehow hide the attach files button if we wanted to but I guess that has to be done on our end since pup is just a js file.