I would suggest that the synagogue of Satan is not merely those who claim to be Jews but are not. it is anyone who allies with such, and whose religion is something public and not of Christ. For Christ is not here to make our life good, our wallet full, while people are gently herded to perdition. He is here to ensure that we do not fear the next life, and we are spared from the final damnation. The world hates Christ for a reason, and the synagogue of satan follows the spirit of the world.
I have not had many sermons on this church. People don't want to face the Jewish issue, but even more they don't want to preach that the righteous in this life will suffer, for the spirit of this age knows that his time is short.
8 “And to the angel of the church in Smyrna write: ‘The words of the first and the last, who died and came to life. 9 “‘I know your tribulation and your poverty (but you are rich) and the slander of those who say that they are Jews and are not, but are a synagogue of Satan. 10 Do not fear what you are about to suffer. Behold, the devil is about to throw some of you into prison, that you may be tested, and for ten days you will have tribulation. Be faithful unto death, and I will give you the crown of life. 11 He who has an ear, let him hear what the Spirit says to the churches. The one who conquers will not be hurt by the second death.’Revelation 2:8-11 ESV