Probably not. You would have a job coming up with a good explanation as to why the Council decided to pick a star deep in hostile geth space instead any of the other billions of stars in empty, unhabited systems though.
If I were writing it, my answer would be that it wasn't a Council project. If I were given a free hand to decide this stuff myself, what I'd do is, the ark would be a skunkworks project taking place around the same time as the Lazarus project, and launching (mostly in secret, but with participants from various citadel-space races) before Shepherd's resurrection.
I mean, Shepherd's ressurection was kept secret, right? So to the rest of the galaxy, this guy who had saved the galaxy and had been warning everyone about the reapers was killed. The set of people who believed him about the reapers would not have been "none", and, well. That would have caused reactions among groups besides the Alliance, the Council, and Cerberus, and those reactions would not be coordinated or in many cases even public.
That's the best way I can see to make it easily compatible with everyone's playthrough of the trilogy. Cured the genophage? Doesn't matter, these Krogan left the Milky Way before the cure, so our Krogan will still be suffering from it (and won't overpopulate Andromeda). Caused the extinction of both the Quarians and the Geth? Doesn't matter, because again, the ark left before that happened. Heck, you could have Shepherd die in the suicide mission, or even survive it and then have an affair with Morinth, and again, it would all be consistent. As long as you weren't going to bring the Rachni along anyway, it could be made consistent with everyone's headcanon.
As for why one would pick that particular star? You could say that non-heretic Geth were involved in the project, or you could say that Quarrians were involved (remember, secret project) and considered the destruction of a solar system in Geth space was a desireable side-effect, or you could say something else.
(But, I'm very much not in charge, so never mind.)