Choice of solder influencing quartz crystal?
Your problem is almost certainly not a direct issue with the solder itself but with the flux in the solder. Many fluxes leave a conductive film on the board which must be removed for circuits such as yours. 

Note that RTC chips try to be extremely low power so as to preserve the battery as much as possible. As such, the biasing of the 32.768KHz crystal for the oscillator is designed to be rather low current (i.e. high impedance circuit). The flux conductivity can sometimes be low enough to completely swamp out and literally short out the oscillator preventing it from working.