What are the rules surrounding the passing of the Darksaber Blade and the leadership of Mandalore?

A "mock duel" would go in against their sense of tradition and code of honor. If a weapon is supposed to be earned you can not do a pretend fight or give a free pass, that would totally defeat the entire point of earning the blade in the first place.

Just like when Mando admitted he didn't kill the horned-beast that baby Yoda saved him from. Gotta earn your stripes.

To expand on the above answer, the obvious follow-on question would be "Since a mock-duel wouldn't be allowed, then why not have an actual match for the blade?" A real fight would likely satisfy honor, and it wouldn't have to be to the death, as Gideon still being alive shows. However, Bo Katan wouldn't use this option either, because even if the fight was legitimate, the story of how she gained the blade would not have the same power. Defeating Moff Gideon and earning the blade is a compelling story, one she believes she needs in order to restore her planet and people. No matter how honorable fighting and winning the blade from Din Djarin would be, that story would not hold the same near-mythic power.

"I hunted down Moff Gideon, who stole the darksaber, and got the blade back" is a story you can tell and rally the Mandalorians with.

"Someone else did it, and then we sparred, and I won" isn't.

And a lie is fragile.

For Bo Katan to base her right to the throne on a lie like that, she'd either have to trust everyone there to keep that lie, or kill everyone present either immediately or in the future. As anyone who knew the truth of what happened would be a vulnerability in her strength forever, politically astute people would even know that knowing that lie makes them a threat to Bo Katan.

Her best bet is probably to send someone else, anonymously, to kill Din Djarin; then kill that person herself. But even that isn't as nearly as good as defeating Moff Gideon herself by a huge amount.

The Darksabre, being the mark of the ruler, is supposed to be wrestled from the hands of the previous ruler; taking from the hands of someone who destroyed the previous thone is a good alternative. Winning it in a duel between two Mandalroians, neither of them rulers, is weak sauce.

I can only offer speculation as I don't know of any official source that explains Bo-Katan's motives during the events in The Mandalorian. In fact, her story hasn't been fully told yet. There might be parts of her backstory that haven't been published yet and that shed more light on her reaction. We don't even know how she lost the Darksaber in the first place, for example.

But based on what we know, I would like to offer a slightly different take than the other answers. Based on Bo-Katan's history my guess is that tradition and appearances aside, this is actually a personal issue for her. Because Bo-Katan never won the saber in the traditional sense in the first place.

Depending on how closely you interpret the rule of having to win the saber from its previous owner, you might say that Maul was the last legitimate owner and everyone after him was not. He challenged Pre Viszla to single combat and claimed both the Darksaber and the Mandalorian throne after defeating and killing him. Maul kept the blade even after he was ousted from Mandalore and eventually kept it on Dathomir along with other artifacts from his past.

Years later (and to make a long story short) it came into the possession of Sabine Wren and thus found its way back into the hands of the Mandalorians.

Sabine was looking for a strong and respected leader to unite the Mandalorian clans against the empire. She believed Bo-Katan to be that leader and offered her the saber. But Bo-Katan didn't share Sabine's view of herself and refused. After her past mistakes, she doubted herself and her ability to lead. She thought she had failed her people and was unworthy to lead them. Sabine ultimately convinced her that she was the best choice and showed her, that she already had the support of the Mandalorian clans even without the saber. Sabine was convinced that she only found the Darksaber so she could pass it on to Bo-Katan.

Only then Bo-Katan accepted and Sabine simply gave her the saber in front of the assembled Mandalorians. And this is where things get tricky. Everyone was absolutely fine with it even without her earning the Darksaber properly. Tradition aside she was accepted as their leader. Or at least that is the way it's depicted in Star Wars Rebels.

Like I said at the beginning, we don't know the full story. We don't know everything that happened between then and the events in The Mandalorian. It could be that her rule wasn't as unanimously accepted as it appeared. Maybe it was disputed by some from the start. Or maybe her leadership was put into question after the events surrounding the Great Purge and the loss of the Darksaber.

Regardless it seems from prior precedent that bending the rules and simply accepting the saber isn't totally out of the question. The situation might have changed insofar as that she has lost so much support, that the other Mandalorians won't accept such a claim to rulership a second time.

But my guess is, that she needs this more for herself. I think she needs to properly lay claim to the Darksaber this time not just to convince the other Mandalorians to follow her but also to prove to herself that she is a worthy leader. She doubted herself before and it stands to reason that she would do so again after not being able to save her people from the purge and also losing the symbol of rulership.

She needs to do this the right way to prove that the trust in her was not misplaced.