Something worth clarifying is whether you have access to a Mac or PC to use Scrivener for these final phases, or are you stuck with just the mobile version? A lot of the above is going to require the full version.
If you do have the full version available, then here is the method I use. This is in response to a different, but very similar, request of having a particular image following the chapter heading. Both an image or an epigraph will have similar mechanical desires though: special formatting, an easy way to see them in the binder, etc.
For iOS-only your options are limited and it’s probably simplest to just type the epigraph in with the formatting you want, and then apply the “Preserve Formatting” markup to that text so the compiler doesn’t change it to look like body text (you’ll need to add that button to your auxiliary toolbar; long tap on any button you don’t want, to change it). It has no concept of “section types”, and what it can do with level based formatting is very simple.
You can still use a separate section just for the quote, like I demonstrated, so that it doesn’t have to be put into some other thing that isn’t really about the quote, but that’s entirely up to you. I prefer to break out elements in the outline so I can see what has what at a high level, and to not have elements attached to things that might move around them (move the first section down to the third, remember to cut and paste the epigraph into what is now the first section).