It seems to me you're over-thinking this. What would happen if you did nothing at all to coordinate? That is, suppose you just have everyone clone the repo, edit the file, and push their result without worrying about what anyone else is doing...
Someone will "just happen" to push first. That push will go through. The next person will get an error saying they're not up to date. So they'll pull, resolve the simple conflict (if there is a conflict, which there may or may not be), push, and be done.
The worst case is that multiple people are resolving conflicts at the same time, and so all but one of those will have to re-resolve; which, granted, could get annoying if everyone's trying to complete this at the same time. But assuming you have a reasonable time window in which this is going to take place, it's likely that won't be a big problem. And in fact this is more or less the point with git - a large number of people can work on the same file(s) concurrently without contrived coordination.
Now, some people would argue that the resulting history is, in their view, "messy" because any time someone had to pull then push, a merge commit is created. Worst case, that's still not worse than what it would look like if everyone added their name on a separate branch. And, it's possible to avoid even that (e.g. by using 'get pull --rebase' when prompted to pull before pushing).