You could try host-based authentication to log in without a password. In this case you don't need a private key for the user. But you still need a host key. Would this be a possible solution for you?
The man page of ssh sais:
Host-based authentication works as follows: If the machine the user logs
in from is listed in /etc/hosts.equiv or /etc/ssh/shosts.equiv on the
remote machine, and the user names are the same on both sides, or if the
files ~/.rhosts or ~/.shosts exist in the user's home directory on the
remote machine and contain a line containing the name of the client
machine and the name of the user on that machine, the user is considered
for login. Additionally, the server must be able to verify the client's
host key (see the description of /etc/ssh/ssh_known_hosts and
~/.ssh/known_hosts, below) for login to be permitted. This authentication method closes security holes due to IP spoofing, DNS spoofing, and
routing spoofing. [Note to the administrator: /etc/hosts.equiv,
~/.rhosts, and the rlogin/rsh protocol in general, are inherently insecure and should be disabled if security is desired.]