Does Jiro/CashRegister...Person ... even need to be a thing for the plot to happen?
Like, assuming everything else being the same, except for Chie being a dude, instead of being a love interest, Cash Register Guy would just be this guy Jiro thinks is cool because he helped Jiro dodge an Emergency Level: Yuudai this one time, but because he (predictably) embarrassed himself in front of him, something along the lines of a pratfall then a "you looked like a good boyfriend and I just wanted to watch", he's too shy and embarrassed to reach out and nab himself a buddy.
That's when we go back to the beginning of the comic, where Jiro half-reclaims his balls and semi-successfully asks Cash Register Girl out on a date, or "interest meeting", and he gets her number, except it's Cash Register GUY, not Girl, and it really isn't a date, but an honest-to-God interest meeting.
Now, Taro sees everything, and thinks that it IS a date, and starts grappling with the non-fact that his dear brother is gay, despite the latter fervently denying romantic intentions.
You can see that this has very quickly turned into straight-up fujobait.
Later, Taisei freaks out over Jiro not wanting to spend the weekend shopping with him, and forces the reason out of him. Jiro blurts out that he's going out with someone, but like, not as a date, and Taisei's laughingly incredulous, same as in canon, then all like "suuuure, it's not a date", and also gets him to tell him with whom he's going out, and Jiro then blurts out that it was that GUY from the market, to his immediate regret. Taisei puts 2+2 together and deduces that Jiro's "date" is a guy. Shocked, but also happy for him finding himself, Taisei whips out his spiel about loving Jiro no matter what or who he does, and Jiro denies that he's dating a dude, and Taisei's goes all puppy dog sad about Jiro not really accepting himself yet, then sends him on his merry way to Harajuku, and frenziedly calls Taro.
So yeah. I guess it could work