I wonder if her uncle, Emperor Karl VI, would be able to be a scary enough strawman to persuade the Russians into going with her? I'm not sure Karl VI would do anything more for his niece than he "did" for Pyotr II, but that doesn't mean that the idea of him "doing" something on Natalia's behalf isn't going to present. OTL, they went with Anna Ivanovna over Ekaterina (who had a child) since they feared that Ekaterina's husband would come to Russia to throw his weight around. They passed over Anna Petrovna's son because they were scared that his father would use Russia to get Holstein back. I'm not sure why Elizabeth Petrovna wasn't considered for empress then already.
Because Peter's descendants were not too popular at the time of the choice and, anyway, Elizabeth (and her sister) had legitimacy issue: they happened to be born out of a wedlock and a lot (mostly bad) could be said about their mother and her reign.
The members of
Supreme Privy Council wanted a compliant pawn and Anne looked as an ideal candidate. Unfortunately for them, they relied too much on their own "legitimacy" while neglecting the objective factors like military backup. Which is strange if one keeps in mind that each of the leading families, Dolgorouki and
Golitsyn, had a rather capable fieldmarshal. To succeed they needed to provide a serious military presence of the loyal troops in the capital but they did not bother and in no time the nobility (aka, the Guards) made it clear to Anne that Supreme Council does not have ...er.... "popular support" and "conditions" are worthless.
Eventually, it took Elizabeth years of a dedicated courting of the Guards AND a very weak regime of Anne Leopoldovna to stage her coup.
And, considering her [Natalia] competition is a widow with no children (Anna Ivanovna),
Not a minus. Plus at least some governing experience.
a child with a claim to the Swedish throne (Pyotr III) and her bastard half aunt (Elizabeth Petrovna), I figure Natalia stands as good a chance as any of them.
Well, yes. It all boils down to who is going to support her. If she spends some quality time drinking with the Guards (and courts some capable military figure able to use them when time arrives) then I'd say, she has a VERY GOOD chance.