The inspector seems way more bothered of Diode not wanting to give her legal name then of her being a female twister.
I would have expected a stronger reactions from him, consider the in universe traditions.
Question to the translator
@MapleMakarov (or who read the raws/another source):
In chapter 3 I was under impression that Die Overdose was her name + family name (contracted to the nickname Diode ), but now her name was state differently. There was a hint to translate the name as that in the previous chapter or it was a completly different thing? The only thing is "Gendo" that translator translate as "limit".