You say that empty pass phrase SSH identity keys are not allowed, but if you're going to hardcode in passwords anyway, perhaps you could simply automate loading the ssh key and then you could avoid having to Expect the ssh login.

   # Run your perl script inside of an ssh-agent(1) context
   $ENV{SSH_ASKPASS} = "/path/to/script/that/echos/password";
   system("ssh-add id_filename </dev/null");
   # Continue with your process
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Otherwise, assuming that you read the docs and are dead set on doing this anyway, I think you're looking for IPC::Open3.