You have to use the FastMail app if you want to send from an email alias. That's the only reason I have for not using the native iOS mail app. Please let me know if you have a workaround!
(Disclaimer: this is undoubtedly true on macOS; I haven't done this sort of thing on iOS in ages.)
Edit: here it is: https://www.fastmail.com/help/clients/iphone-aliases.html I'm going to give it a try now. Thanks!