Can I instigate collaboration outside of my current research team, or is that bad etiquette?
You should not start working on an independent project without telling your current supervisor. After all, he acquired the funding which pays your salary, so he needs to be on board with everything you do that takes up a significant amount of time. (Of course, I am not talking about asking for his permission every time you review a paper.)

Then again, as a postdoc (as compared to Ph.D. students), you are expected to start spreading your wings and collaborating with people outside your supervisor's immediate group. So chances are that your supervisor will be supportive, especially since you write that you have some time to kill.

But ask first; it's possible that your supervisor knows something that needs to be done before your project "really" starts.

This question is related (full disclosure: I answered there): Should postdocs work only on the projects they are hired for, or can they work with other people in the same research group on other projects?