Try saving assembly as part, and then save the (assembly) part as STEP (or other neutral format).
You can use the "Defeature" option found in the "Tools" menu which will remove all the internal parts & fill holes of your choice.
I've never run into that type of problem with STEP, IGES or Parasolid files. I have with ACIS (.sat) files. You might try Parasolid or IGES files. Which one to use will depend upon what works best with the receiving system as well as what you can get out of SolidWorks. Parasolid is the native format for SolidWorks, as well as for NX and SolidEdge, so it will export best from SolidWorks and will most likely import best into NX and SolidEdge.