Paul Calhoun sent me some sample code that I massaged to get it to produce the spreadsheet that I wanted. I don't know what it was that he'd done that I hadn't, but, for now, I think this is the heart of the solution, just harnessing an OutputStream instead of either a FileOutputStream or ByteArrayOutputStream.
// The Faces Context global object provides access to the servlet environment via the external content
var extCont = facesContext.getExternalContext();
// The servlet's response object provides control to the response object
var pageResponse = extCont.getResponse();
//Get the output stream to stream binary data
var pageOutput = pageResponse.getOutputStream();
// Set the content type and headers
pageResponse.setContentType("application/x-ms-excel");
pageResponse.setHeader("Cache-Control", "no-cache");
pageResponse.setHeader("Content-Disposition","inline; filename=" + fileName);
//Write the output, flush the buffer and close the stream
wb.write(pageOutput);
pageOutput.flush();
pageOutput.close();
// Terminate the request processing lifecycle.
facesContext.responseComplete();
I will gladly provide help if someone else encounters this issue and, hopefully, by the time someone asks, I will understand more of what was different that worked....