In medieval Europe the Church, of course, permitted marriages to end only under the rarest of circumstances. But Jewish law allows divorces very easily (especially for the man). If a Jewish man (maybe in Poland or the HRE) in the middle ages wishes to divorce would he be able to do so under Jewish law or would civil authorities insist on a more strict standard. Did the governments leave this sort of thing entirely up to the local Jewish authorities or did they insist on some standards across religious lines.
Divorce was actually quite common in medieval European Jewish communities, to the extent that, according to scholars like Avraham Grossman, various rabbinic rulings were put into effect from both sides of the equation to reduce the number of divorces: on the one hand, the cherem (excommunication) of Rabbeinu Gershom in the 10-11c which stated that men cannot divorce their wives without the wife's consent, and on the other hand the rabbinic rulings in the 13-14c which made it harder for a woman to unilaterally demand a divorce.
But your question isn't really about that- it's about whether Jews were allowed to divorce in a Christian society in which divorce was not permitted as a general rule. And the answer is yes because they were, in many way, not living in a Christian society. Jews were generally considered outside the typical power structure of the Christian lands in which they lived, generally their own communal entity under direct control of the local ruler (or, in the case of the Jews of England pre-1290, literally the property of the king). These Jews were generally organized into their own kehilot, or communities, led by the kahal, or communal leadership. These were highly organized communal structures which relied on Jewish law as arbitrated by both local and distant rabbis in order to create communal charters and rules, collect taxes, create social service networks, etc. They could be highly sophisticated- and highly independent. (I talk a bit more about this, though with a focus on a later era, here.)
The taxes were important because while some of these taxes were for the needs of the community itself, they were even more importantly intended to raise the money that would allow the ruler of the place in which the kehila was located to remain. (I talk more about this dynamic here.) If the sum wasn't paid- or if the sum was increased and became out of the community's reach- the Jews in the kehila would have to leave the area and find a new place to settle, which could be difficult to impossible depending on the current political situation. However- as long as the sum was paid, the community was essentially autonomous in terms of the way it governed itself. For the most part, the edicts of the Catholic Church were not necessarily relevant, and the Church and local government probably never knew about many of the births, marriages, divorces and deaths that occurred in the kehila.
So, essentially, the fact that the Church didn't officially approve of divorce didn't matter- Jews did their things their own way and ruled themselves autonomously within the Christian lands in which they settled. As long as the rulers left them alone, that is. The question really ended up being what happened once this autonomy disintegrated, which (to make a long story short) really kicked into gear with with emancipation in the 19th century. In this post I discuss how, in fact, the emancipation- or rather dissolution of the autonomy- of the French Jewish community in the time of Napoleon meant that, when French rabbis were asked whether divorce was permitted, they essentially said "since divorce is prohibited in France, Jewish law prohibits divorce in France." That is not something that the residents of truly autonomous Jewish communities in medieval times would have said, despite the Catholic rules regarding divorce remaining the same.