[Further to my previous message...]
[When starting a non-interactive non-login shell from an ssh session...]

 >   What bash does is source /etc/bashrc and ~/.bashrc--it evidently

 >   treats ssh like rsh and does a little more than is documented even for

 >   rsh (search for rsh in the bash man page for details).
So another, simpler, system-wide solution is to set your $PATH in

/etc/bashrc. However, this is not documented in the bash man page (which

is why I came up with the more complicated solution, as I hadn't

realised this behaviour at that stage). So I'm not sure whether it is

standard/will occur on all systems, etc.
If you're like me and like to have interactive stuff in bashrc and login

stuff in profile, you can arrange your files like this (very similar to

before, but ~/.bashrc needn't source /etc/profile now, nor do its check

with $- and grep, as /etc/bashrc does that and avoids setting $PS1).
    In /etc/profile
      # System-wide .profile for sh(1)

      PATH=...

      export PATH

      # Source /etc/bashrc for interactive shells.

      [ -n "$PS1" -a -r /etc/bashrc ] && source /etc/bashrc

      ... stuff for login shells (more environment mostly)
    In /etc/bashrc
      # System-wide .bashrc file for interactive bash(1) shells.

      # If the shell isn't actually interactive it is an ssh session, and

      # we want to source etc/profile instead.

      if [ -z "$PS1" ] ; then

        [ -r /etc/profile ] && source /etc/profile

        return

      fi

      ... stuff for interactive shells (aliases and terminal stuff mostly)
    In ~/.profile
      # For login shells (interactive or not).

      # For interactive non-login shells see ~/.bashrc.

      # For non-interactive non-login ssh shells see ~/.bashrc.

      # For other non-interactive non-login shells see $BASH_ENV.

      # At present this is sourced by ~/.bashrc for the ssh case, so

      # all ssh sessions act like login sessions.

      PATH=...

      export PATH

...
[ message continues ]