Template Button Relocation

yes. If this is something you must have, I would post in marketplace - the composer toolbar isn’t readily customizable. The gear is usually default for new buttons. I managed to do it with my Image Alignment and Grid component, so you could look at that for clues. I don’t remember offhand how I achieved it though.

It is possible you may need a plugin to do more composer toolbar manipulation- there is code in the formatting toolbar plugin that controls buttons, as well as some tips on controlling them I think.