As discussed in the comments, your only option is a Proxy. Everything else (Like mod_rewrite / Alias) will do a header redirect, and expose the target address to the user.
However, Apache's proxying doesn't seem to be that complicated. The most basic example is
ProxyPass /mirror/foo/ http://backend.example.com/
Note that a proxy will of course double your traffic, because every byte requested will have to be fetched from the remote machine.
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