I love how Arianne NEVER imply anything or bring up rumors relating to Myrcellaās allegedĀ illegitimacyĀ in her POV. Arianne wants her crowned as queen of Westeros because an older sister must inherit before her younger brother and because Myrcella is much smarter than what she should be for her age and would make a great queen.
To be fair, it wouldn't serve Arianne's personal political goals to bring up Myrcella's alleged illegitimacy, as it would make her claim to the Throne nonexistent and Arianne's ploy redundant. If she did find out, however, I don't think she'd care that much. In Dorne, there doesn't seem to be as much stigma against children born out of wedlock if the Sand Snakes are any indication. They're treated as Princesses of Dorne in spirit, if not in name. Likewise, while it's not entirely unusual in Westeros for heirs to be raised along 'bastard' siblings or offspring in their respective House seats, it is quite unusual to have the mother living alongside them. If the 'bastards' of noblemen are allowed to live with their Lord fathers, their mothers are typically dismissed back to their homes. However, Oberyn lives with Ellaria Sand in Sunspear and continues to father children with her. They seem to live in a partnership in all except the legal sense, and no one seems to really give a shit.
It's not brought up at all in either Doran, Arianne, or Areo Hotah's pov chapters, nor is the inclusion of the Sand Snakes et al at Sunspear. Though we have limited pov chapters with Doran, it's not mentioned that any of his leal lords have an issue with their presence either. The women seem to enjoy quite a bit of freedom and luxury, and then Nymeria is sent as an envoy to King's Landing to sit on the Council; a position that is unheard for bastards, especially female ones. Once again, the issue of 'bastardry' isn't brought up, and it truly seems to be a non-issue in Dorne. Which would be fitting considering that Dorne relies on inheritance based on age rather than gender, and one of the reasons 'bastards' are so stigmatized is to police women's sexual lives and limit autonomy. If the only power a daughter can offer is getting married, then 'remaining pure' is the primary way you ensure the best 'deal' for her, because her future husband will want unquestioned heirs. Meanwhile, if your daughter is going to serve as heir, then any heirs will go through her and questioning their inheritance is a moot point. With less limits as to women's power, it makes sense that discrimination against illegitimate children would be lessened as well. And again, none of the Dornishmen we've seen thus far seem to give an iota of a shit about illegitimacy. The Sand Snakes feel perfectly comfortable sassing Doran, and he barely even stops them.
Like you said, Arianne decides to install Myrcella because she is very smart and already has forged strong relationships in Dorne. If Myrcella were to be Queen, then Dorne would be in a much more auspicious position with King's Landing, especially considering Trystane would be the King Consort and House Martell would be Myrcella's in-laws. She would be much less likely to start a war with her favorite in-laws, and she would perhaps lend Dorne favorable trade conditions.
(Edit: I wrote this ask before reading Arianneās chapters in the TWOW sample and some of my thoughts have changed.)
