Hello Michiel,
TrillSpanners live as outside staff objects. So you have to tell lilypond in this case that it does not. Then simply specify a Y-offset as you wanted, relative to the staff centre.
\once \override TrillSpanner.bound-details.right.padding = #2.0
\once \override TrillSpanner.outside-staff-priority = ##f
\once \override TrillSpanner.Y-offset = #1
I leave the spacing of the D as an exercise for the reader.