Classic "I have no choice but to participate in this completely unreasonable, unnecessary and stupid duel to prove my worthy, otherwise I'll be shaming my family and ancestors!" bullshit.
I have mixed feelings, on one hand this is typical Japanese bullshit, on the other hand, maybe the real logic is that since Satou is basically an orphan she don't want to ruin Makino relationship with her family, which is why she suggested this, after all if they win toxic mom will have no choice but to accept their relationship.
One reason I suspect that is the reason is because toxic mom is only using the "abnormal environment she was raised in" as argument, not that Makino is gay, so with Satou proving she's a good influence she won't be able to do anything anymore.
This is stil bullshit of course, but it's a very japanese thing that makes sense, they have issues with orphans and such after all, not to mention shame and so on, Japan can be really backwards sometimes, so it's not as much bullshit as most examples, I guess it feels worse than it really is because most of the times this kind of thing makes absolutely no sense even in-universe and it's just excuse to force a duel.