the original uses it as a highschool girl thing
the formal apology would be "my apologies" and the more casual one is "sorry", however she is using the more casual one while also skipping a consonant
using something like "my bad" would not really fit since she is apologising even if its in a sloppy manner
think it can be considered odd even in japanese since it sounds like something you'd hear from a student trying to get a store clerk's attention or for something between friends
my take on it is that she uses this extremely informal approach since he saved her before and seemed nice, also the fact that he has his 2 servants with him so she wants to be friendly
In that case, I can see "Sowy" as the best fit.
"My bad" can be a genuine apology, it is all about tone.
I can see where you are coming from with the difference between "My apologies" and "Sorry" depending on the region and the English being spoken, whether it is American English (U.S.), British English (U.K.), or International English. In my experience in the U.S., I don't see "Apologies." used as much as the informal, but it does come up. I have definitely seen it more in International English and British English.
Words are always changing, "sorry" used to be considered almost always formal until more recently, in the past 200 years or so, there is some debate on whether it is mid-18th or mid-19th Century. Heck, the definition of "Apology" being a synonym for "Sorry" was not even created until Shakespeare decided to do so. Words are weird, especially in a language as convoluted and warped as English.
I appreciate it if you read this far. I am currently procrastinating doing my own English assignments and correcting other people's papers. It was nice to write casually for a second. I hope my ramble was interesting to you, at least.
