You can use custom projections in PostGIS by inserting rows into the spatial_ref_sys table (no need to know the EPSG-assigned SRID, use whatever you like as long as it does not collide with already assigned ones). This, however, involves knowing both the WKT and the PROJ.4 form of the projection string: [1] is a table of mappings between WKT and PROJ.4 expressions, [2] is a way to automate the conversion between the two using OGR.

[1] http://cfconventions.org/wkt-proj-4.html

[2] https://gist.github.com/hrwgc/4435920