Bridge Rectifier Fails When AC Inputs are Shorted
You should not be trying to run your power converter by wiring into the circuit across the doorbell switch. Instead you should be wiring directly across the output of the doorbell transformer.

As you have it now you are powering your converter through the impedance of the existing doorbell. Even you describe that as a solenoid which is an inductive load. When the door bell button is released there is a good chance of an inductive spike of high voltage that is taking out the diodes in your bridge. 

This is a diagram of how the wiring for the circuit should be wired instead of tapping across the existing door bell button. Even with this corrected design it would be advisable to check if there are voltage spikes when the switch is released and design in appropriate high voltage spike clamping into your adapter circuit.