@caitsith That's because dryly using the imperative'd be too rude for Japanese standards considering he is running a business(I know the series isn't set in Japan, but these customs are hard to get rid of when you write). Even when you are a complete ass, shopkeepers and staff still politely ask you to get out, even if their tone is anything but polite.
As for using 『行きな』, it's a speech pattern I only hear from some older people, but it has the same meaning as using dictionary form, the 『な』 in there is roughly the same thing as putting an "ok" at the end.
Btw, it seems I was right about the kanji, although it can also be written with 『他所』(my word processor actually recommends this kanji over the other one when I type "yoso", in fact) for the same meaning, and the jisho page for it has the "going elsewhere" meaning explicitly: https://jisho.org/search/%E4%BB%96%E6%89%80%E8%A1%8C%E3%81%8D.
That said, both meanings are acceptable either way: https://dictionary.goo.ne.jp/jn/227852/meaning/m0u/(1.よそへ行くこと 2.外出するときの衣服や持ち物), so sounds like it's up for context, and I think the interpretation she should go away makes more sense