One question: you add some extra padding between the two variants with an \override; is there a way to do the same thing for modifying the padding between two PianoStaff groups (but not change the staff-staff spacing within each of them)?
Absolutely. Just one catch - it has to be done differently. For all the gory details, I'm going to suggest you have a look through chapter 4 of the Notation Reference, titled simply Spacing issues. It's at
http://lilypond.org/doc/v2.25/Documentation/notation/spacing-issues There's a lot in there, but the main thing to recognize and be aware of is that different spacing settings get set in different places. Some can go in a few different places (such as the staff-staff padding I added in the example code), while others can only go in the paper block. And as it happens, the setting you are asking about is one that has to go into the paper block. If you already have a paper block (for example setting margins or indents, that kind of thing) you can just add this in. But basically what you'll need is something along the lines of:
\paper {
system-system-spacing.padding = #15
}
Here, system-system-spacing is the setting you want. And each setting has sub-settings - basic-distance, minimum-distance, padding, and stretchability. The docs I linked to above have all the details on what each of those mean and how to go about changing them. The number is the space measured in staff spaces. It'll probably take a bit of trial and error to find the spacing you want, but it should be fairly straight forward.