Normally, a person looses around 40ml of blood per period with people having 450 periods on average in their life. This means that on average a person would loose 18l of blood from menstruation alone. With the average human having between 4.5l and 5.7l, you'd only need to sacrifice 4 people to the menstruation gods to never have to shed blood for that ever again. ~~In my experience, it works best when sacrificing transphobes or terfs.~~
As a quick side note: I hate it when someone refers to it as a "pseudo-period". The hormonal cycle is very very real, I can assure you that.
It might be better to pursue it and eventually be honest with them. As you learn more about their flaws, they will stop being an unreachable figure, and start to become a person you understand. At the same time, you don't need to hide your feelings as much (they might pick up on them, as like me, you probably suck at being sneaky). They'll feel respected and understood, especially if you make it clear that you'll accept a rejection.
That said, only try this if you can spend a lot of time with them. If you can't see each other very often, it would be hard for even a super charismatic person to form a meaningful bond. If they give a rude rejection to someone being so honest and vulnerable, you deserved better. If they give a kind rejection, you can move on and might be able to stay as friends. All potential options are better than the alternative, so be brave!
This advice (that I have never tried) should be taken with a handful of salt. I just looked at what didn’t work for me in the past, and assumed my anxieties then were totally counter productive. If someone was obsessed with me, I'd want them to behave this way. I want others to show me respect as an equal person, not hate themselves while seeing me as perfect.