There is certainly some controversy around this battle, to which I have an admittedly small amount of knowledge on, though will share what I can. I believe the Alliance, and Landais, acted at times on their own accord, ignoring Jones’ orders. While the ship didn’t really desert, it didn’t really engage directly and acted more on the periphery (at times even hitting the Bonhomme Richard with shots, presumably by accident), while it shot at Royal Navy ships, including the Serapis.
I do think Landais was trying to help Jones, and the American cause, just not with firey enthusiasm, more just enough not to look bad. I think his ship actually ended up under Jones after the batte, an ironic series of events.
If anyone can weigh in with other details or more information, it is certainly welcomed.
Also thought it might be interesting to read through this letter from Jones to Benjamin Franklin, explaining some of the battle: https://founders.archives.gov/documents/Franklin/01-30-02-0366