Large current draw at each cycle
This is perfectly normal and is something of a downside of the capacitor input power supply, but they all do it. 

Generally (unless you model the parasitics correctly) spice models of this sort of thing are a bit worse then the charging pulses are in reality, because real transformers have leakage inductance and resistance and real grids have both inductance and resistance. 

The old school solution is a choke input supply, but it adds quite a lot of weight and needs a minimum current load to avoid the voltage shooting up, this was popular in the vacuum tube era where lots of volts at low current was the usual case. The modern way if a PFC circuit, but it adds a lot of complexity. 

How much ripple can you tolerate? Reducing the size of the cap increases ripple, but raises the conduction angle, so less cap is better in that respect. 

What almost everyone does is just to massively over size the diodes and size the transformer for the actual VA requirement (which will be quite a lot bigger then the average power), and carefully consider the cap ripple current.