Why does the output voltage after full bridge drop so many?
I believe the problem is the reverse current of the diode (current which flows through the diode if it is reversed biased). For this diode, this current is about 2uA, thus during the positive half cycle, the source current will flow in the forward direction through D2 and part of it back to the source through D3 (reverse current).

Probably if you use ideal diodes in your simulation, the problem should be solved. Or as you already suggested, just increase the source current.