You could use the ZCTA postcode as a SHP file as a background layer using a local WMS server (e.g. Geoserver), as you don't wish to have this in the level of detail. I'd advise you use PostgreSQL with PostGIS and load the ZCTA shapefile into that, and then connect that table to Geoserver.
The background layer will look something like this: ZCTA shapefile served as a WMS background
And you'll need USA Admin 0. Both files are available on the Tiger website. I'd advise styling these as white with a grey outline, you could use AtlasStyler to create the SLD.
You could then load up your custom shapefile as a foreground layer (datasource) to perform the analysis.