begin any social program cuts in districts whose Representatives voted for those cuts; let them lead by example.

Allow those districts whose Representatives voted for cuts to lead the way. Let these districts show that they can get along with less government intervention before applying the cuts to the rest of the nation.
If the Congress wants to assess social welfare program reform, then let those districts that don't want it be the primary subjects - it will be much easier to see which cuts work and which do not by having some districts as 'experimental' and others as 'control'. It will also be less costly both to remove and restore benefits.
Those Representatives who have voted for cuts have made their districts de facto volunteers for such research.