Hello Tom,
The Great Synagogue was really on of London's finest. Established in 1620, it was rebuilt on the same site of Duke's Place, Aldgate in 1790. It was sadly destroyed in the WWII.
Although Esther is, of course, simply a biblical name, combined with Barnet it is a fairly obvious anglicized Jewish name.
Esther's father's name illustrates some of complexities of Jewish naming practice.
Yehuda was his main name (Judah in English). Yehuda's father, Jacob, likened his son to a lion at some point. The Hebrew for lion is ariyeh, which used as forename and often shortened to Ari. The Yiddish word for lion is Laib/Leib/Leyb (pronounced 'libe').
The father had the double Hebrew name Yehuda Ariyeh and the Yiddish nickname Laib.
It his likely that his father's forename was Baruch (the same origin as Presdient Obama's forename), which was commonly anglicized to Barnet.
Now to Philip; his Hebrew name - name as the shem kodesh (holy name) - was used in Hebrew documents, e.g. birth, marriage, divorce docs etc., as well as to call him up in the synagogue to read a blessing.
Ha-Levi is Hebrew for 'the Levite', which means that he and his male ancestors were supposedly descended from the tribe of Levi. In Jewish hierarchy, this was second only to kohanim (Cohens = the priests). A web search will reveal thousands of links for further reading.
To my knowledge, the Jews in Prussia were not subjected to the Russian-style pogroms at that time. The most likely reason for emigration was simply financial - in many parts of the Prussian empire, it was difficult for Jews to earn a living - or military service in the Prussian army. All men of 21 were obliged to do seven years of military service. I think that was the cause of my ancestors' emigration in the 1870s.
The GS register records the surname spelled Grizmish. A more common spelling is Gryzmisch - the 'sch' being germanized. However, you'll also see Grzymisch or Grzymish. I cannot recognise a Jewish orogin. Maybe one of our Polish experts can recognise some Polish root.
I'll let you absorb this.
Justin