I just made up my mind and went forward installing the plugin (thanks @angus). It seems to work quite well, but I encounter an unpleasant bug.
When I click the Babble button, all the page’s content moves a few pixels to the right. Actually, clicking the Babble button makes my browser’s scrollbar (Firefox 55.0.3) disappear, however long the page may be.
Someone pointed out it made the footer disappear when he clicked Babble’s button – I can confirm this happens on my forum too.
I’m using a slightly modified default theme.
Anyone had this fixed?
edit OK, I found it in babble-shoutbox-init.js.es6, line 43: