Chris,
How about this? Is this more the spacing you were looking for?
I did it by explicitly setting the location of the first and last system of the interlude measures. I'm not sure if you CAN do it any other way at the moment.
Here's the code I used (in the InterludeMusic variable) which you should be able to apply (by trial-and-error, unfortunately) to the other instance on page 4:
%<---------- SNIP -----------
...
% Interlude
\mark \markup { \hspace #25 "Musical Interlude" }
\overrideProperty Score.NonMusicalPaperColumn.line-break-system-details #'((Y-offset . 65))
\VerseOneUnison
b8. a16 b8 c | d8. d16 b8 g | d'8. d16 b8 g | a8. r16 d,4^> | \break
\overrideProperty Score.NonMusicalPaperColumn.line-break-system-details #'((Y-offset . 96))
g8 a g fs | e r a4^> | fs8 a c b | g4. r8 | R2 | \bar ".|:-||"
% Verse 4
...
%<---------- SNIP -----------
Oh, and I moved the first measure from the interlude's bottom system to the end of the second one (which you can change back, of course :-) I just thought it looked less crowded). Also, I've learned that putting in manual \breaks can make a big difference to how Lily uses the \overridePropertys.
Regards,
Abraham