There is a specific term for Jews that hate Jews and/or Israel. They are “antijews”.
Importantly, they are not inherently “antisemitic”, as many of them are very pro-Arab, a Semitic people, even though in the common parlance, antisemitism is almost exclusively used to describe anti-Jewish and anti-Israel things.
Judaism itself has a very nebulous definition. In some ways religious, to varying degrees; in other ways ethnically Semitic, but so heavily blended with other races that they have lost most of these ethnic traits.
The two major Jewish ethnic blends are the Ashkenazim and the Sephardim. Ashkenazim, or “Germanics” have German Jewish cultural and geographical origins, while Sephardim, or “Hispanics”, are so named denoting their Spanish/Portuguese Jewish cultural and geographic origins.
Then there is cultural Judaism, which has some commonalities, such as the cyclical understanding of history, being in a perpetual minority status, and the effects of both extended ghettoization and integration with other cultures.
So, as I said, Jews have a nebulous definition. But “antijews”, who often call themselves “Jewish”, are very clear about the Jews and Israel they hate. They hate all of it and them, and seek their destruction.
It is mindless to the extreme. Perhaps it reflects their self hatred, but extrapolated to an entire people. It may have some elements of racism, classism, it is often very atheist in character. At the same time it is often mindlessly idealistic, passionately embracing any crackpot philosophy as long as it promises perfection, and also hates Jews.
They are the “antijews”, and that label should be applied to them. Any pretense of their calling themselves Jewish is just ridiculous.
Just call them Kapos.
Last year, I read Paul Johnson's History of the Jews. He usually called such people "non-Jew Jews". They were self-hating Jews who eschewed everything that made the Jews unique. He included Jews like Marx and Trotsky when he used that description, so I guess an old Bolshevik like Bernie belongs on the list, too.