If both you and your gf live together with the baby, from it's birth until the end of the year, then either of you can claim the child, and the child tax credit, on your tax return. It does not matter what address you use.
However, the statement "I can't claim the baby at her address or it could mess with her taxes" indicates that either you or her or both of you are planning on doing something that is not allowed.
As others have said. either one of you can claim the child, but not both. You are allowed to decide between you who that will be. Only if you can't agree do the IRS rules govern. The it's the parent with the highest income (assuming you both live with the baby an equal amount of time).
Neither of you have to be Head of Household (HoH) to claim the child. But nobody can be HoH, unless they are claiming a qualifying dependent. You do not have to be "on the lease" to be HoH, you only have to have paid most of the household expenses, which usually includes the rent.
