How to specify the private SSH-key to use when executing shell command on Git

Something like this should work (suggested by orip):

ssh-agent bash -c 'ssh-add /somewhere/yourkey; git clone git@github.com:user/project.git'

if you prefer subshells, you could try the following (though it is more fragile):

ssh-agent $(ssh-add /somewhere/yourkey; git clone git@github.com:user/project.git)

Git will invoke SSH which will find its agent by environment variable; this will, in turn, have the key loaded.

Alternatively, setting HOME may also do the trick, provided you are willing to setup a directory that contains only a .ssh directory as HOME; this may either contain an identity.pub, or a config file setting IdentityFile.