Yes, once he has lung metastases and they talk of 'floating' cells, chemo is the one option: but a big field these days.
Neither radiotherapy nor surgery would be useful, except to deal with local issues (eg bone pain). Although there a variety of hormone preparations, they all work on the same principle, and if the cells are no longer receptive, that will apply to all.
If the cancer cells are no longer hormone-sensitive, this suggests that they are becoming 'undifferentiated'; they are 'cancer cells' no longer acting like 'prostate cancer cells', and don't necessarily affect PSA levels.
Also, once the cancer is metastasising, progress is more usefully assessed by scans, signs and symptoms.
The next step is probably considering which chemo regime would be appropriate to the stage he is at - and there may be research options.
-- Andrew --
"I intend to live forever, or die trying" - Groucho Marx
Sorry to see this, Mary. They may suggest adding Enzalutimide or Abiraterone?
"Life can only be understood backwards; but it must be lived forwards." Soren Kierkegaard