Okay.
First, player suicide would be the player killing themselves. Not the player having the character kill themselves. You are talking about CHARACTER suicide, not PLAYER suicide. That is a very, very important distinction to make.
Next, I think you might want to review something. If a player has no desire to continue to play the same character, why is it necessary for a character to die in order for the player to switch roles? Someone ending their own life is a fairly heavy topic you are treating with a weird sort of levity here. The motivating factor for the character to stop adventuring is coming from the player's desire to play something else-- i.e. the motivating factor is something outside their own reality. Having the character commit suicide is unnecessary and pointless.
So this tells me not enough time was spent in creating the character. To you and the player it is just a piece of paper with a bunch of numbers written on it via the guide the rulebook indicated. It isn't a character or else a stronger conceptualization of this individuals motivations and drives would be clear and the idea of the character killing themselves would be utterly unthinkable.
That being said, if the player has no desire to play a character, there is no need to force them. The character can just become an NPC and step out of the adventuring party with some fairly conceivable excuse (he made enough money during the adventures to be able to afford to live a less dangerous life or a family member wrote to him and asked him to come back because a parent died or someone was gravely ill, etc.). There is no need for a character to die nor any retconning to be done for a player to switch roles over to another they are more comfortable with in 98% of cases. Unless all the characters are truly trapped in some limbo that precludes the possibility of encountering other people or someone leaving the presence of the others of their own volition, then allowing a player to swap character roles is not impossible.
Now... you aren't really presenting a good argument against this "Blood Hunter" class. Saying that something "isn&t in the world and doesn't exist" is an empty and asinine excuse that doesn't hold up to scrutiny. Because, if you are going to be completely honest the very moment you come across a monster that you hadn't thought up before but like, you will retcon it into existing in the world as though it always had in numbers large enough to propagate itself... if you come across a magic item you want to use but hadn't previously introduced into the world, you will feel free to add it to the world and quite possibly in a way that indicates it has existed in the world for centuries. Basically you just arbitrarily add things to the world whenever it strikes you fancy, and so you really aren't being honest in saying you can't introduce something into the world as always having been there because it doesn't exist in the world-- because this is something you surely do and will continue to do going forward. When you use that excuse, what you are actually saying is "I am saying no arbitrarily simply because I said so, go screw yourself." In which case they are completely correct in being upset by this behavior on your part.
Which isn't to say you are incorrect in disallowing the class to be played. But in that case you right when you said "I have nothing wrong with the class"-- well, you were wrong regardless because that is absolutely atrocious use of the English language and doesn't actually have a meaning, but I gather you meant "I have nothing against the class" which surely cannot possibly be the case if you are arbitrarily disallowing it. There is clearly something about the class that makes you not want it to be allowed. And there are plenty of valid reasons why-- from the mechanics requiring too much work to the theme of the class not matching the tone of the game you want to run to not feeling prepared or experienced enough to really judge the mechanics.... to simply being suspicious of anything that is homebrewed because even if everything looks on the up-and-up on first glance, it is quite possible you are missing the loophole that allows it to be vastly overpowered and ruin everyone else's fun.
But you need to be honest with both yourself and your player first. If you deal honestly with people, you'll find many of these problems are not so difficult to resolve.
