Use `sudo -s -H -u git-auto-deploy`.
`sudo` by itself does not change the value of `HOME`, but does so when `-H` is used. This is why you're getting that "Permission denied" error. The user can't access `root`'s home directory (and shouldn't be able to either).
The `-s` option makes `sudo` start whatever shell is configured as the login shell for the specified user (i.e. no need to explicitly run `/bin/bash`).