To reduce the spacing, you can do
\new ChordNames \with {
chordChanges = ##t
voltaOnThisStaff = ##t
\override VoltaBracket #'minimum-space =#0
} { ... }
This looks like it should work, and LP accepts it OK, but it doesn't
have any effect. The volta brackets are still much too high. I can't
see any difference in output appearance between the versions with and
without the "\override VoltaBracket #'minimum-space = #0" line. Below
is my current LP source in case you'd like to try and see if it really
works for you.