Is it permissible to attend a non-frum event, such as a wedding or bar/bat mitzvah of a family member or friend?
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The Gemara speaks harshly about those are invited to a Seudaat Mitzva (according to R' Mutzafi this means weddings or Beritot) and don't attent. However, HaRav Ovadia Yosef Shelit"a writes (Yabia Omer vol. 4 Yoreh Deah 19) that if there are going to be "Inshi Delo Maali" e.g. non-frum people it's better not to go. (I heard this from Rav Meir Eliyahu Shelit"a).
This sounds like the sort of question that you really need to discuss with your Rabbi, since there are lots of different issues in play, and how they interact could vary a great deal, depending on the circumstances and on the people involved.
I'll enumerate some of the major issues that could come up. Note that I'm not saying that all of these are necessarily important issues in all situations.
This is just my guess on things from what I've seen.
There can be a lot of issues going on in each case, so it's worth taking an honest assessment of the full situation, and talking with a rabbi who's both knowledgeable and understanding. If both you and the event host demonstrate genuine caring and communication, that can help a lot of things too.
I think the overarching principles here, in descending priority, are:
A.) We can't violate strict Halacha.
Otherwise, my guess is to respect C, D, and E above, so long as they don't violate A, B, and E. Examples that could potentially violate A, B, and E, and are thus more problematic:
If the hosts are making an effort to accommodate you, that can be a reason to attend (and not violate C, D, and E), but it has to be balanced with the above concerns.
One more note: Orthodox rabbis (even some fairly right-wing ones) are used to getting these questions. Otzar HaPoskim has the case of a BT woman attending her brother's wedding, at which she'll be expected to kiss him. (I guess it's just a huggy-kissy kind of family.) Answer: it's not so much Halachically prohibited as frowned upon, which gives some leniency. But better to explain the situation to him if you can.