I think in this sense, "goat" is referring to a scapegoat. The singer refers back to days a "long time ago" where men were lucky and enjoyed wine, but now "hell is here" and people no longer enjoy things. Instead, we are all looking for someone/something to blame our problems on, which in this case seems to be God, or religion in general. The title then makes sense, as "Hallelujah" is often used in a religious setting, so a Hallelujah Goat would be using religion as a scapegoat to blame problems upon.
I think in this sense, "goat" is referring to a scapegoat. The singer refers back to days a "long time ago" where men were lucky and enjoyed wine, but now "hell is here" and people no longer enjoy things. Instead, we are all looking for someone/something to blame our problems on, which in this case seems to be God, or religion in general. The title then makes sense, as "Hallelujah" is often used in a religious setting, so a Hallelujah Goat would be using religion as a scapegoat to blame problems upon.