I've stumbled up on the answer to my own question. Either of the following should work:
\set restNumberThreshold = #1000 % Some large number \override MultiMeasureRestNumber #'stencil = ##f
The first is a bit hackish and not very robust (it sets the number of rests required to display the number extremely high), but the second may have unintended consequences in some older versions of Lilypond.
Here is the final result: