Assuming that one can ascertain that food cooked by an irreligious Jew is kosher, is the food still prohibited due to Bishul Akum? Does this also apply in a case of need, such as one visiting an irreligious relative, where it may be embarrassing and inconvenient to procure other food?
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Halachically Speaking (7:2) has a great overview of this question:
From http://www.shemayisrael.com/parsha/halacha/Volume_7_Issue_2.pdf, which also has all the references in footnotes
Rabbi Elyashiv Zatzal in Kovaitz Teshuvos 3:115 says that it is possible that the Gezaira of Chasnus applies also to a Mechalel Shabbos. Even though technically one can marry a non observant Jew, the Gezaira was made to keep us separate.
Maharam Schick Orach Chaim 281 also says that anywhere the Chachomim made a Gezeira on non-Jews it applies to Jews that are Mumrim.