What's the most plausible way to get a Jewish-Majority state (either independent nation or a state in a federal union) in North America with a POD of 1800?
Former is hard, latter is fairly easy. Like DanMcCollum said, and as I've explored the idea in the past (but haven't posted the TL, which is very much a work in progress). A Western state or prairie province could easily get a large wave of Jewish immigration in the late 19th century, and the tiny base populations involved could make the place majority Jewish without too much trouble.
I'm not sure that Jewish-majority is possibly with a PoD of 1800 - maybe Napoleon tries to encourage Jews to move somewhere? But even though we technically have 3 years before the Louisiana Purchase, in practice France is probably still going to do it, and 3 years isn't a lot of time, and I don't see the US reacting super well to France encouraging Jews to immigrate to an American holding. It's hard to do Haiti or one of the sugar islands, if the Caribbean even counts as North America.
I've also played with the idea of immigration to Alaska by Jews under Russia, but I just can't find any reason why the tzars would allow, let alone encourage, mass Jewish migration.
It's not Zionism if they don't go to Zion.
They had plenty of antiSemitism in the 19th century but no nucleus of new Zion forming then.
...Zionism, per se, starts in the mid 19th century. There was a strong faction of the World Zionist Congress that wanted a Jewish state anywhere - famously, Argentina (both the pampas and Patagonia) and "Uganda" (actually part of Kenya today) were proposed and considered seriously. Herzl, in fact, was notionally in favor of such a plan.
Despite the poor name, Zionism has more to do with the awakening of a Jewish national identity, in response to national awakenings across Europe, which together with scientific racism saw attempts by Jews in the West to assimilate to be increasingly rebuffed. The conclusion that the Jewish nation needed a Jewish [national] state came quickly, by way of the doctrine of nation-states which would see, for example, agitation by Czechs in Austria-Hungary for their own state, stuff like that.
I think the POD has to be more of a pull than a push; some rather liberal Jews in Europe catch the whole Utopian settlement wave
Interesting idea, though probably religious rather than liberal: utopianism was mostly driven by religion, and the liberal Jews of Europe were fiercely secular, if not outright anti-religious. But definitely a cool direction.
Hebrew wouldn't be used at all in your hypothetical state. Hebrew had been a dead language for centuries by the 19th century, and Israel only reseructed Hebrew for the sake of Arabic speaking Sephardi Jews ( so the could communicate with Yiddish speaking Ashkenazim Jews). As the united states at the time restricted immigration from Africa and the middle east, there will be no substantial population of Sephardi Jews and thus is no need for the main language to be any thing other than yiddish
The revival of Hebrew has nothing to do with Sephardi and Mizrahi Jews; it is a wholesomely Ashkenazi and Zionist endeavor, aimed replacing the "language of the Diaspora" with the "language of Zion", swapping the old Jew, oppressed and weak, for the New Jew, strong and self-directing. Hebrew was promoted in Palestine from the first Zionist settlements in the 1870s, despite major Mizrahi and Sephardi immigration only coming after 1948, as various Middle Eastern countries saw pogroms in response to the formation of Israel.
Also, a lot of the assimilated, secularized Jews that spearheaded Zionism didn't (or wouldn't) speak Yiddish, preferring French, German, English, etc. Though probably most of the people who actually settled did speak Yiddish. For reference, the official language of the first World Zionist Congresses was German, and all notes were taken in that language, and delegates were urged to speak in it.
Also: a pet peeve of mine: you should have written Ashkenazi Jews or Ashkenazim; Ashkenazim Jews is like writing "Germans doctors"