Contrary to that link, I'm pretty sure that in this case 漢 is meant to be read as "Otoko." (man) It's an example of one type of 当て字 (ateji), the type where a character is used for a native Japanese word due to its meaning instead of for the typical reading. The character (which can also in the "kan" meaning mean China as well as "man") is sometimes used for "man" in cases where 男 would just seem like a little too generic "man." 漢 as "otoko" would carry more of a connotation of "man's man," "real man," "guy," or something like that, which is what's being implied here. It's definitely an "unofficial" reading, not approved of by the Japanese Ministry of Education, but you'll see it a lot in manga and such (generally with furigana above it.)
See for reference:
here or
here or a long discussion
here.
Since they describe the genre as "漢はやらなきゃいけない時がある" "otoko" just sounds better here. (Though some people might read it differently.)
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