First step, of course, was to try Pageant. No dice. Then I tried choosing "native ssh" - that just made IntelliJ hang (with no way to cancel - really nice GUI there, guys).
Then the shoe dropped; maybe it pretends to be a "real" ssh client? One short check of my Windows home directory later I'd dropped my
id_rsaprivate key file in %USERPROFILE%/.ssh, and IntelliJ happily picked it up.