I think it's within the realm of reason that Sai would have picked up on birthdays in the meantime (when he's recognizing history questions, he's answering Edo Period questions, if my memory serves). And I really don't know the Edo Period customs.... To be honest, I wouldn't change the story just to fit with my memory of how things work in the Heian Period.
Death anniversaries have been observed since the Nara Period, so birthdays aren't that much of a stretch. And Sai might not know when he was born, exactly--I've never seen anyone talk about when exactly they were born in Heian diaries: just when their children, or the children of important people (like the emperor) were born. (And that's usually for astrology, like when one guy said that he wasn't exactly sure what hour his son was born, because it was cloudy and they couldn't check the stars.)
I think, since it'd be the old calendar, Torajiro's birthday is 5/5 (in general, old calendar dates are earlier than the modern conversion--that's the lunar-solar calendar, after all).
But the story works! I don't think you need to change it, unless you'd really really want to. And, honestly, it's not even like that many people in Japan would be able to answer the question of whether Edo Period people celebrated birthdays (and even fewer for the Heian Period).
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Death anniversaries have been observed since the Nara Period, so birthdays aren't that much of a stretch. And Sai might not know when he was born, exactly--I've never seen anyone talk about when exactly they were born in Heian diaries: just when their children, or the children of important people (like the emperor) were born. (And that's usually for astrology, like when one guy said that he wasn't exactly sure what hour his son was born, because it was cloudy and they couldn't check the stars.)
I think, since it'd be the old calendar, Torajiro's birthday is 5/5 (in general, old calendar dates are earlier than the modern conversion--that's the lunar-solar calendar, after all).
But the story works! I don't think you need to change it, unless you'd really really want to. And, honestly, it's not even like that many people in Japan would be able to answer the question of whether Edo Period people celebrated birthdays (and even fewer for the Heian Period).