I searched the Dutch website
www.wiewaswie.nl/en and found no births fro a Charles or Saul.
I tried variant spellings of Berlin, Berlyn and Berlijn.
There were Berlijn familes registering births in Utrecht and Nieuwveen, around 1810-1820.
The family in Utrecht would appear to be Jewish, going by the forenames (rebekka, Salomon, Moses, Machiel etc).
As for passenger lists - none were kept (or recorded) of travel between Europe and the UK. Unless the relevant boat was also coming from further afield.