I've been using an iPhone for almost exactly 2 years now and I didn't realized until like a month ago that you can change the default password manager. Unless Keepass just didn't tap into the API (which would surprise me), but Enpass works like a charm and I felt like an idiot for manually going into the app and copy-pasting this whole time.
Settings > Passwords > Password Options
And then set "allow filling from" to your password manager