Rikuo and Shinako both had plenty of opportunities to tell Haru and Rou that they had started going out. They both got busted right outside their homes by the people who were persuing them romantically, that just made the whole thing more painful.
I'm not 100-percent sure of if that is grammatical. I just have a feeling that the writer should have used which instead of that.
The sentence is not correct as written. It is possible that the writer incorrectly used "that" instead of "which", as you suggest. Another possibility is that the writer means "that" as a demonstrative not relative pronoun.
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The sentence is not correct as written. It is possible that the writer incorrectly used "that" instead of "which", as you suggest. Another possibility is that the writer means "that" as a demonstrative not relative pronoun. In other words, the writer means:
They both got busted right outside their homes by the people who were pursuing them romantically. That [i.e. the fact that