"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.