The Laws of the Amazons

"In the event that a woman warrior is defeated by an outsider,
if her opponent is a woman, she must give her the Kiss of Death
and kill her.
However . . .
If her opponent is a man, she must make him her husband."

If Mousse beat up Shampoo, would Shampoo have to marry Mousse?

 Almost certainly not. There are no laws stating what happens when a woman is beaten by someone of the same village. If it was the same as regarding the outsiders, then obviously the law would say it applies to everyone because it wouldn't be sensible to have it as two separate laws. So, it is very likely that men of the village don't have to marry any female they beat. Also if the law did apply to anyone, the women wouldn't be able to fight with others at all, because someone would loose, so someone would have to die, or get married every time a female has a fight.
Kiss of Death
Amazons have very big pride.  So losing to an outsider, is worse
shame than death.  The kiss of death is a promise, to chase you
to the end of Earth, and kill!
Shampoo has given the Kiss of Death to both Ranma-chan and
Akane in graphic novel number 3
What would Mousse have to do to get Shampoo to love him?
Cologne has said (V5.6) that the women of her village want to marry a
strong man, and to Shampoo Ranma is the strongest young man in the
area, and also very strong compared to everyone in the world.
So, Ranma is the best available man to her, and so Mousse would have
to be shown to be better than Ranma for Shampoo to consider
marrying Mousse - though Shampoo does love Ranma, which'll cause
some problems. Also, he'd have to stop mistaking other objects/people
for Shampoo - which pissed her off when they were children and so she
rejected him, and still does because he hasn't learnt yet. If Ranma married
Akane, then Mousse would have an easier time, though Shampoo would not
necessarily give on Ranma
immediately, but if he was the strongest bachelor in the village he
would be best suited for marrying
Shampoo.