Shalom.
I note that your question is specifically NOT what does Halachah/Jewish Law say about this matter. You ask about minhag/custom.
Answering you from that perspective, I would point out that minhagim/customs differ in various communities. Clearly, in a very devout or observant community, the custom would follow the Halachah, and few or no persons would get a haircut as you describe the situation. On the other hand, in a community where the bulk of the members were Reform, this would be a matter of personal decision, and some might refrain, while others (perhaps the majority) would have the haricut.
The question you ask presupposes a universal response, and that just isn't likely.
The situation is not unlike the concern about naming a child: in the Ashkenazic Jewish community, it would be considered against the prevailing custom/minhag to name after a living relative, while in the Sephardic Jewish community, it would be considered in keeping with the prevailing custom/minhag to name after a living relative. Minhagim/customs differ from community to community, place to place, and tme to time.
If you are asking whether you should go against a prevailing minhag/custom, I would suggest that the presumption should be no, you should not violate minhag/custom, at least not without a very good reason. This same idea is expressed well in the English adage, 'When in Rome, do as the Romans.' That works, insofar as the 'Romans' we are talking about your those in the community you come from.
I hope that this has helped to answer your concern.
Rabbi Joe Blair
Answered by: Rabbi Joseph Blair