If a Jew could avoid forced labor or even death by giving up their religion (or even pretending to), did millions of Jews bravely choose not to do that and rather endure through the ridiculous oppression?
Or was it that they'd be persecuted anyway even if they said they'd give up their faith
No.
The Third Reich defined Jewishness via ancestry in the Nuremberg Laws. Religious observance, or the lack of it, was irrelevant for the Nazi racial system. A German could have been the most enthusiastic Christian convert or an atheist from a family of the most non-observant Jews to have ever walked the earth, and it would not have mattered. The only cover Christian conversion imparted was that long-term converts often possessed social networks among the gentile majority that provided a modicum of protection. But such protection was often fleeting and relatively powerless.
In short, no, in fact there were some who had converted a generation or more before the 1930s who were still persecuted and killed. Though the Nazis did rely on religious antisemitism, and relied heavily on compliance and collaboration of Christians based on a particularly Christian type of religious antisemitism, the Nazis themselves engaged in antisemitism that was primarily racial. Their definition of who was a Jew did not rely on self identification, religiosity, participation in a Jewish community, or Jewish law regarding Jewishness. Rather their identification was purely racial, a Jew was anyone with one or more Jewish grandparent, this being derived from the American "one drop" rule in determining race for cases surrounding miscegenation and segregation. Jews were categorized into several racial categories "full Jews" and "Mischlinge" or mixed Jews. In most of the places the Nazis conquered, and the areas where most Jews lived, intermarriage was uncommon, but it was somewhat common in Germany and other areas of Western Europe, in certain areas the status of mischlinge was applied and was sometimes somewhat protective, but in Eastern Europe where the majority of Jews lived, and where the majority were killed anyone with any Jewish grandparents were considered Jews, and could be killed or enslaved with impunity. A full Jew was defined as someone with three or more Jewish grandparents, and/or were involved in a Jewish community. A first degree mischling was someone not involved in a Jewish community or had two Jewish grandparents. Those who were not involved in a Jewish community and had one Jewish grandparent were called mischlinge of the second degree.
However, this is not to say that people didn't attempt to survive by "passing" as Christians. This was particularly true of very young children, who were often put into hiding with Christian families or in Catholic orphanages. This was also more common for girls and women than it was for men and boys because Jews are routinely circumcised, which was not true of most European non-Jews at the time. Baptismal or Christian marriage records and other documents could be forged to protect Jews, sometimes entire families. But if the Nazis suspected that the papers were forged, they could still be deported to a killing center or killed immediately.
In areas such as Poland, Lithuania, Ukraine, and Belarus, where the majority of European Jews lived, Jews tended to be religious, or come from religious families, and most were Yiddish speaking. Hiding a Jewish identity meant not only obtaining papers proving ones status as a non-Jew, it often meant learning Christian prayers and traditions to better pass, and passing was impossible for those who only spoke Yiddish or spoke with a heavy accent, as well as those who had features that were stereotypically associated with Jews. Obtaining false papers was also expensive, so for the vast majority of Jews in Europe hiding was not an option.
Christian, and in particular Catholic organizations such as orphanages and monasteries were in a particularly strong position to provide hiding for young children who in many ways were easier to hide and explain away than adults. With the centrality of conversion, many churches had motive to try to keep children from returning to Jewish communities after the war, and task forces were eventually created to help reunite the hidden children with any remaining family or at least reconnect them with their Judaism.
https://www.ushmm.org/collections/bibliography/christianity-and-the-holocaust
https://www.ushmm.org/collections/bibliography/hidden-children
https://encyclopedia.ushmm.org/content/en/article/the-nuremberg-race-laws