It's probably worth noting that even in Death Watch not all the Mandalorians held the same beliefs. Vizla and a chunk of his followers seemed relatively content that Maul take his place by right of strength, while Bo Katan and her splinter group considered Maul an 'outsider' and refused to follow him. Presumably this would also reflect how they would view Lux.

Pre Vizla basically seemed to be a might makes right all the way type of person, nothing else really mattered, the strongest should be in charge and define what it is to be Death Watch, and only Death Watch are true Mandalorians. Basically making being Mandalorian more like being a pirate or a gang member. It's a justification for being a violent person more than a genuine respect for tradition. Lux can be adopted as a Mandalorian simply because Pre Vizla decides he wants to and is in charge.

Bo Katan seemed to think of it as a matter of being raised in, or at least following, the old warrior creed of the Mandalorians before their civil war. An outsider who proves stronger than the current leader doesn't have the right to redefine the Mandalorians even if they won a formal challenge. She probably wouldn't have accepted Lux as a Mandalorian unless he did actually adopt a lot of their customs, because for her the traditions and customs are the part that matters.