Your Fitbit may be an Alta HR, rather than an Alta. I have the same situation you do. The PCB in the HR is different, and in the center has a very small flat-flex cable to the HR sensor below. If you are very careful, you can fold back a small piece of tape holding the flex flex in place (look closely, it is very hard to see), and then you can gently disconnect the HR sensor. The connector lifts straight up, perpendicular to the surface of the PCB. After you have disconnected, you can remove the two screws and slide out the screen, PCB, and battery assembly as a single unit. The HR sensor will remain behind in the case. I would recommend using a plastic tweezers for this process rather than metal, as if the PCB is powered you could damage it in the process.
I took some photos, but I don’t believe I can post them here.