As the .cookie file is always created anew on bitcoind start and removed once bitconid is stopped, I don't know if chgrp would set the group permanently for the file. I wouldn't expect it to, as it's always a new file with new content.
Can't you just recreate the hardlink and change group permissions each time bitcoind starts?
how can the hard-linking to the other users home extend the file permission from a file it couldn't read in the first place?
I was thinking something like this:
chgrp btcCookie .cookie
chmod 640 .cookie
ln /home/bitcoin/.bitcoin/.cookie /home/anotheruser/.cookie
And "anotheruser" needs to be in group "btcCookie". I usually just edit
/etc/group for that.