When to use bucketing vs. other methods of grouping data

(originally posted to StatsiQ but I think this is the better location.)
Hello all, I'm new to Qualtrics. I am looking for advice about how to use bucketing or other related data grouping methods so that I can group responses to narrow down my analysis. Let me give a simple example. My survey asks respondents to identify their gender by checking all that apply among the following options: woman, man, transgender, non-binary, other, prefer not to say. I would like to group these into Cis Woman, Cis Man, and Other (due to very few responses from non-cisgendered people). I am concerned that bucketing will not work because the bucketed variable will only assign them to their first selected answer. This will not be a problem for someone who checked "man": this response will properly be bucketed into "man." But for a trans man who checked "man" AND "transgender", will there be a problem? Will he also get grouped with "man", even if he should be grouped into "other"?
Do experienced users have suggestions for other ways to go about grouping responses? I have several other responses that might be grouped in a similar way, and manually tagging them all seems inefficient and clunky. Thank you for your help.