Writing a commentary to a scientific journal without one's advisor I started as a research assistant in a new lab 10 weeks ago. I have a six-month fixed-term contract, which was supposed to be extended to a PhD project in 2017. However, my supervisors have been mean to me (I don't quite understand why) and therefore I do not plan to stay in the group.

I have one unfinished paper (from the previous lab) which I plan to submit as soon as possible. The situation is quite clear: it was work that was started elsewhere, which I complete on my own time, and hence my new supervisors will not be co-authors.

I have been advocating against a certain practice in my field for about one year. I have received good comments about it. I would like to try to publish a short commentary in a leading science journal about it. However, I am not sure how to go about doing it. I would like to be the sole author, and not even discuss the work with my new supervisors. If I do it on my own time, is there potential for problems regarding authorship? It appears this particular lab has a policy that the supervisors co-author everything. Anyway, the usual protocols (e.g. Vancouver) suggest that without contributing they should be not. From a legal perspective I guess there are no problems, since the work (1) concerns a very general topic, (2) constitutes only of the idea, and does not need any resources from the university, and (3) I did it on my own time.

One thing I am also wondering is what to put as an affiliation. Should I put my present lab? Since it is the place where I work... Actually, doing that will have a positive effect to the lab's funding prospects if it is mentioned as an affiliation. On the other hand, I guess it could be used somehow against me in the worse case.

What should I do?

Please note that this is about a commentary, not original research. I think that makes this question different from many others.
I see no reason not to send this as a single author paper. 

I would, however, mention it as a courtesy to any new adviser you may acquire. If they are reasonable, they will not require being authors, but they may have input that could be useful to you, and it would certainly be useful for them to know about the paper.