I lack the appropriate toolbox to know, nor do I have expertise in this particular field. (I'd have made this an answer if I did.)
But it seems to me that you need to be careful with the translation of camera 3. You know the translation of 3 wrt 2, but that translation is described in the coordinate system of camera 2. That leaves you with a disconnect, because you want it all in the camera 1 coordinate system. So you need to back the 2-->3 translation through the camera 2 rotation from 1 to 2.
Logically, it might be better if you did everything with respect to camera 2 anyway, as the camera in common. Then you don't need to back things up to camera 1.
An alternative approach is to pair camera 1 and 2, then camera 1 and 3. That leaves camera 1 as the common camera, so it is much simpler to unravel things.
Just some vague thoughts, that might be helpful.
